@inproceedings{RH:86,
author = {R. Hamming},
booktitle = {Bell Communications Research Colloquium Seminar},
note = {Transcripted by James F. Kaiser. (Later version of
the talk at
\url{https://www.youtube.com/watch?v=a1zDuOPkMSw})},
organization = {Bell Communications Research, Morristown, NJ},
pages = {483},
title = {You and your research},
year = {1986},
url = {http://motion.me.ucsb.edu/RHamming-YouAndYourResearch-
1986.pdf},
}
@article{SB:1922,
author = {Banach, Stefan},
journal = {Fundamenta Mathematicae},
number = {1},
pages = {133--181},
title = {Sur les op{\'e}rations dans les ensembles abstraits
et leur application aux {\'e}quations int{\'e}grales},
volume = {3},
year = {1922},
doi = {10.4064/fm-3-1-133-181},
}
@book{EZ:86,
author = {E. Zeidler},
publisher = {Springer},
title = {Nonlinear Functional Analysis and Its Applications.
I: Fixed Point Theorems},
year = {1986},
isbn = {978-0-387-90914-1},
}
@book{MAK-WAK:01,
author = {M. A. Khamsi and W. A. Kirk},
publisher = {John Wiley \& Sons},
title = {An Introduction to Metric Spaces and Fixed Point
Theory},
year = {2001},
isbn = {9781118031322},
}
@book{AG-JD:03,
author = {A. Granas and J. Dugundji},
publisher = {Springer},
title = {Fixed Point Theory},
year = {2003},
doi = {10.1007/978-0-387-21593-8},
isbn = {978-1-4419-1805-5},
}
@book{VB:07,
author = {V. Berinde},
publisher = {Springer},
title = {Iterative Approximation of Fixed Points},
year = {2007},
isbn = {3-540-72233-5},
}
@article{BPD:61,
author = {B. P. Demidovi\v{c}},
journal = {Uspekhi Matematicheskikh Nauk},
number = {3(99)},
pages = {216},
title = {Dissipativity of a nonlinear system of differential
equations},
volume = {16},
year = {1961},
}
@book{NNK:63,
author = {N. N. Krasovski\u\i},
note = {Translation of the 1959 edition in Russian by J. L.
Brenner},
publisher = {Stanford University Press},
title = {Stability of Motion. Applications of Lyapunov's
Second Method to Differential Systems and Equations
with Delay},
year = {1963},
}
@article{CAD-HH:72,
author = {C. A. Desoer and H. Haneda},
journal = {IEEE Transactions on Circuit Theory},
number = {5},
pages = {480-486},
title = {The measure of a matrix as a tool to analyze computer
algorithms for circuit analysis},
volume = {19},
year = {1972},
doi = {10.1109/TCT.1972.1083507},
}
@article{LC-DG:76,
author = {L. {Chua} and D. {Green}},
journal = {IEEE Transactions on Circuits and Systems},
number = {6},
pages = {355-379},
title = {A qualitative analysis of the behavior of dynamic
nonlinear networks: {S}tability of autonomous
networks},
volume = {23},
year = {1976},
abstract = {Several theorems are presented which predict in a
qualitative manner the behavior of a large class of
dynamic nonlinear networks containing coupled and
multiterminal resistors, inductors, and capacitors. A
very general and rather surprising result is
presented which guarantees that most autonomous and
nonautonomous dynamic nonlinear active networks of
practical interest have no finite "forward" escape
time solutions. In the case of autonomous networks,
sufficient conditions are given which guarantee that
the solution waveforms possess various forms of
stability properties. The concepts of eventual
passivity and eventual strict passivity are invoked
to guarantee that all solution waveforms are bounded
and eventually uniformly bounded, respectively. The
properties of reciprocity and monotonicity (local
passivity) are invoked to guarantee that all
solutions are completely stable. The further
imposition of a growth condition guarantees that all
solutions will converge to a globally asymptotically
stable equilibrium point. In this case, the magnitude
of all solutions is shown to be bounded between two
exponential waveforms for all timet > 0. An algorithm
is presented which computes for the maximum
"transient decay" time constant associated with the
upper bounding exponential. The main features of the
majority of the theorems presented in this paper are
that their hypotheses are simple and easily
verifiable-often by inspection. The hypotheses are of
two types: first, very general conditions on the
network state equations and second, conditions on the
individual element characteristics and their
interconnections. The hypotheses and proofs of the
latter type of theorems depend heavily upon the
graphtheoretic results of an earlier paper [14] and
involve solely the examination of the global nature
of each element's constitutive relation and the
verification of a topological "loop-cutset"
conditions.},
doi = {10.1109/TCS.1976.1084228},
}
@article{MV:78,
author = {M. Vidyasagar},
journal = {Journal of Mathematical Analysis and Applications},
number = {1},
pages = {90-103},
title = {On matrix measures and convex {{Liapunov}} functions},
volume = {62},
year = {1978},
abstract = {In this paper, we extend the concept of the measure
of a matrix to encompass a measure induced by an
arbitrary convex positive definite function. It is
shown that this modified matrix measure has most of
the properties of the usual matrix measure, and that
many of the known applications of the usual matrix
measure can therefore be carried over to the modified
matrix measure. These applications include deriving
conditions for a mapping to be a diffeomorphism on
Rn, and estimating the solution errors that result
when a nonlinear network is approximated by a
piecewise linear network. We also develop a
connection between matrix measures and {Liapunov}
functions. Specifically, we show that if V is a
convex positive definite function and A is a Hurwitz
matrix, then mu V(A) < 0, if and only if V is a
{Liapunov} function for the system x... = Ax. This
linking up between matrix measures and {Liapunov}
functions leads to some results on the existence of a
“common” matrix measure μV(·) such that μV(Ai)
< 0 for each of a given set of matrices A1,…, Am.
Finally, we also give some results for matrices with
nonnegative off-diagonal terms.},
doi = {10.1016/0022-247X(78)90221-4},
}
@article{CWW-LOC:95b,
author = {C. W. Wu and L. O. Chua},
journal = {IEEE Transactions on Circuits and Systems~I:
Fundamental Theory and Applications},
pages = {430-447},
title = {Synchronization in an array of linearly coupled
dynamical systems},
volume = {42},
year = {1995},
abstract = {In this paper, we extend the results in [1994, same
authors, Int J Bifurcation and Chaos] and give
sufficient conditions for an array of linearly
coupled systems to synchronize, A typical result
states that the array will synchronize if the nonzero
eigenvalues of the coupling matrix have real parts
that are negative enough. In particular, we show that
the intuitive idea that strong enough mutual
diffusive coupling will synchronize an array of
identical cells is true in general. Sufficient
conditionsfor synchronization for several coupling
configurations will be considered. For coupling that
leaves the array decoupled at the synchronized state,
the cells each follow their natural uncoupled
dynamics at the synchronized state. We illustrate
this with an array of chaotic oscillators. Extensions
of these results to general coupling are discussed.},
doi = {10.1109/81.404047},
}
@article{YF-TGK:96,
author = {Y. Fang and T. G. Kincaid},
journal = {IEEE Transactions on Neural Networks},
number = {4},
pages = {996-1006},
title = {Stability analysis of dynamical neural networks},
volume = {7},
year = {1996},
abstract = {In this paper, we use the matrix measure technique to
study the stability of dynamical neural networks.
Testable conditions for global exponential stability
of nonlinear dynamical systems and dynamical neural
networks are given. It shows how a few well-known
results can be unified and generalized in a
straightforward way. Local exponential stability of a
class of dynamical neural networks is also studied;
we point out that the local exponential stability of
any equilibrium point of dynamical neural networks is
equivalent to the stability of the linearized system
around that equilibrium point. From this, some
well-known and new sufficient conditions for local
exponential stability of neural networks are
obtained.},
doi = {10.1109/72.508941},
}
@article{WL-JJES:98,
author = {W. Lohmiller and J.-J. E. Slotine},
journal = {Automatica},
number = {6},
pages = {683--696},
title = {On contraction analysis for non-linear systems},
volume = {34},
year = {1998},
abstract = {This paper derives new results in non-linear system
analysis using methods inspired from fluid mechanics
and differential geometry. Based on a differential
analysis of convergence, these results may be viewed
as generalizing the classical Krasovskii theorem,
and, more loosely, linear eigenvalue analysis. A
central feature is that convergence and limit
behavior are in a sense treated separately, leading
to significant conceptual simplifications. The
approach is illustrated by controller and observer
designs for simple physical examples.},
doi = {10.1016/S0005-1098(98)00019-3},
}
@article{WL-JJES:00,
author = {W. Lohmiller and J.-J. E. Slotine},
journal = {IEEE Transactions on Automatic Control},
number = {5},
pages = {984-989},
title = {Control system design for mechanical systems using
contraction theory},
volume = {45},
year = {2000},
doi = {10.1109/9.855568},
}
@article{AP-AP-NVDW-HN:04,
author = {Pavlov, A. and Pogromsky, A. and {Van de Wouw}, N. and
Nijmeijer, H.},
journal = {Systems \& Control Letters},
number = {3-4},
pages = {257--261},
title = {Convergent dynamics, a tribute to {B}oris {P}avlovich
{D}emidovich},
volume = {52},
year = {2004},
doi = {10.1016/j.sysconle.2004.02.003},
}
@article{WW-JJES:05,
author = {Wang, W. and Slotine, J. J.},
journal = {Biological Cybernetics},
number = {1},
pages = {38--53},
title = {On partial contraction analysis for coupled nonlinear
oscillators},
volume = {92},
year = {2005},
doi = {10.1007/s00422-004-0527-x},
}
@article{QCP-JJES:07,
author = {Q. C. Pham and J.-J. E. Slotine},
journal = {Neural Networks},
number = {1},
pages = {62--77},
title = {Stable concurrent synchronization in dynamic system
networks},
volume = {20},
year = {2007},
abstract = {In a network of dynamical systems, concurrent
synchronization is a regime where multiple groups of
fully synchronized elements coexist. In the brain,
concurrent synchronization may occur at several
scales, with multiple rhythms interacting and
functional assemblies combining neural oscillators of
many different types. Mathematically, stable
concurrent synchronization corresponds to convergence
to a flow-invariant linear subspace of the global
state space. We derive a general condition for such
convergence to occur globally and exponentially. We
also show that, under mild conditions, global
convergence to a concurrently synchronized regime is
preserved under basic system combinations such as
negative feedback or hierarchies, so that stable
concurrently synchronized aggregates of arbitrary
size can be constructed. Robustnesss of stable
concurrent synchronization to variations in
individual dynamics is also quantified. Simple
applications of these results to classical questions
in systems neuroscience and robotics are discussed.},
doi = {10.1016/j.neunet.2006.07.008},
}
@article{GR-MDB-EDS:10a,
author = {G. Russo and M. {Di~Bernardo} and E. D. Sontag},
journal = {PLoS Computational Biology},
number = {4},
pages = {e1000739},
title = {Global entrainment of transcriptional systems to
periodic inputs},
volume = {6},
year = {2010},
doi = {10.1371/journal.pcbi.1000739},
}
@article{GR-MDB-EDS:13,
author = {G. {Russo} and M. {Di~Bernardo} and E. D. {Sontag}},
journal = {IEEE Transactions on Automatic Control},
number = {5},
pages = {1328-1331},
title = {A Contraction Approach to the Hierarchical Analysis
and Design of Networked Systems},
volume = {58},
year = {2013},
abstract = {This brief is concerned with the stability of
continuous-time networked systems. Using contraction
theory, a result is established on the network
structure and the properties of the individual
component subsystems and their couplings to ensure
the overall contractivity of the entire network.
Specifically, it is shown that a contraction property
on a reduced-order matrix that quantifies the
interconnection structure, coupled with
contractivity/expansion estimates on the individual
component subsystems, suffices to ensure that
trajectories of the overall system converge towards
each other.},
doi = {10.1109/TAC.2012.2223355},
}
@article{FF-RS:14,
author = {F. Forni and R. Sepulchre},
journal = {IEEE Transactions on Automatic Control},
number = {3},
pages = {614--628},
title = {A differential {Lyapunov} framework for contraction
analysis},
volume = {59},
year = {2014},
abstract = {Lyapunov's second theorem is an essential tool for
stability analysis of differential equations. The
paper provides an analog theorem for incremental
stability analysis by lifting the Lyapunov function
to the tangent bundle. The Lyapunov function endows
the state-space with a Finsler structure. Incremental
stability is inferred from infinitesimal contraction
of the Finsler metrics through integration along
solutions curves.},
doi = {10.1109/TAC.2013.2285771},
}
@article{ZA-EDS:14,
author = {Z. Aminzare and E. D. Sontag},
journal = {IEEE Transactions on Network Science and Engineering},
number = {2},
pages = {91-106},
title = {Synchronization of Diffusively-Connected Nonlinear
Systems: {R}esults Based on Contractions with Respect
to General Norms},
volume = {1},
year = {2014},
doi = {10.1109/TNSE.2015.2395075},
}
@article{IRM-JJES:17,
author = {I. R. {Manchester} and J.-J. E. Slotine},
journal = {IEEE Transactions on Automatic Control},
number = {6},
pages = {3046-3053},
title = {Control Contraction Metrics: Convex and Intrinsic
Criteria for Nonlinear Feedback Design},
volume = {62},
year = {2017},
abstract = {We introduce the concept of a control contraction
metric, extending contraction analysis to
constructive nonlinear control design. We derive
sufficient conditions for exponential sta-
bilizability of all trajectories of a nonlinear
control system. The con- ditions have a simple
geometrical interpretation, can be written as a
convex feasibility problem, and are invariant under
coordinate changes. We show that these conditions are
necessary and suffi- cient for feedback linearizable
systems and also derive novel con- vex criteria for
exponential stabilization of a nonlinear submanifold
of state space. We illustrate the benefits of
convexity by construct- ing a controller for an
unstable polynomial system that combines local
optimality and global stability, using a metric found
via sum- of-squares programming.},
doi = {10.1109/TAC.2017.2668380},
}
@article{SJ-PCV-FB:19q,
author = {S. Jafarpour and P. Cisneros-Velarde and F. Bullo},
journal = {IEEE Transactions on Automatic Control},
number = {3},
pages = {1285-1300},
title = {Weak and Semi-Contraction for Network Systems and
Diffusively-Coupled Oscillators},
volume = {67},
year = {2022},
abstract = {We develop two generalizations of contraction theory,
namely, semi-contraction and weak-contraction theory.
First, using the notion of semi-norm, we propose a
geometric framework for semi-contraction theory. We
introduce matrix semi-measures and characterize their
properties. We show that the spectral abscissa of a
matrix is the infimum over weighted semi-measures.
For dynamical systems, we use the semi-measure of
their Jacobian to characterize the contractivity
properties of their trajectories. Second, for weakly
contracting systems, we prove a dichotomy for the
asymptotic behavior of their trajectories and novel
sufficient conditions for convergence to an
equilibrium. Third, we show that every trajectory of
a doubly-contracting system, i.e., a system that is
both weakly and semi-contracting, converges to an
equilibrium point. Finally, we apply our results to
various important network systems including affine
averaging and affine flow systems, continuous-time
distributed primal-dual algorithms, and networks of
diffusively-coupled dynamical systems. For
diffusively-coupled systems, the semi-contraction
theory leads to a sufficient condition for
synchronization that is sharper, in general, than
previously-known tests.},
doi = {10.1109/TAC.2021.3073096},
}
@article{CW-RP-MM-JJES:20,
author = {C. Wu and R. Pines and M. Margaliot and
J.-J. E. Slotine},
journal = {IEEE Transactions on Automatic Control},
title = {Generalization of the multiplicative and additive
compounds of square matrices and contraction in the
{Hausdorff} dimension},
year = {2022},
doi = {10.1109/TAC.2022.3162547},
}
@article{AD-SJ-FB:20o,
author = {A. Davydov and S. Jafarpour and F. Bullo},
journal = {IEEE Transactions on Automatic Control},
number = {12},
pages = {6667-6681},
title = {{Non-Euclidean} Contraction Theory for Robust
Nonlinear Stability},
volume = {67},
year = {2022},
abstract = {We study necessary and sufficient conditions for
contraction and incremental stability of dynamical
systems with respect to non-Euclidean norms. First,
we introduce weak pairings as a framework to study
contractivity with respect to arbitrary norms, and
characterize their properties. We introduce and study
the sign and max pairings for the $\ell_1$ and
$\ell_\infty$ norms, respectively. Using weak
pairings, we establish five equivalent
characterizations for contraction, including the
one-sided Lipschitz condition for the vector field as
well as logarithmic norm and Demidovich conditions
for the corresponding Jacobian. Third, we extend our
contraction framework in two directions: we prove
equivalences for contraction of continuous vector
fields and we formalize the weaker notion of
equilibrium contraction, which ensures exponential
convergence to an equilibrium. Finally, as an
application, we provide (i) incremental
input-to-state stability and finite input-state gain
properties for contracting systems, and (ii) a
general theorem about the Lipschitz interconnection
of contracting systems, whereby the Hurwitzness of a
gain matrix implies the contractivity of the
interconnected system.},
doi = {10.1109/TAC.2022.3183966},
}
@article{DA-EDS:03,
author = {D. Angeli and E. D. Sontag},
journal = {IEEE Transactions on Automatic Control},
number = {10},
pages = {1684--1698},
title = {Monotone Control Systems},
volume = {48},
year = {2003},
doi = {10.1109/TAC.2003.817920},
}
@incollection{MWH-HLS:05,
author = {M. W. Hirsch and H. L. Smith},
booktitle = {Handbook of Differential Equations: Ordinary
Differential Equations},
editor = {A. Canada and P. Drabek and A. Fonda},
pages = {239--357},
publisher = {Elsevier},
title = {Monotone Dynamical Systems},
volume = {2},
year = {2005},
}
@article{EDS:07,
author = {E. D. Sontag},
journal = {Systems and Synthetic Biology},
number = {2},
pages = {59--87},
title = {Monotone and near-monotone biochemical networks},
volume = {1},
year = {2007},
abstract = {Monotone subsystems have appealing properties as
components of larger networks, since they exhibit
robust dynamical stability and predictability of
responses to perturbations. This suggests that
natural biological systems may have evolved to be, if
not monotone, at least close to monotone in the sense
of being decomposable into a ``small'' number of
monotone components, In addition, recent research has
shown that much insight can be attained from
decomposing networks into monotone subsystems and the
analysis of the resulting interconnections using
tools from control theory. This paper provides an
expository introduction to monotone systems and their
interconnections, describing the basic concepts and
some of the main mathematical results in a largely
informal fashion.},
doi = {10.1007/s11693-007-9005-9},
}
@article{GC:17,
author = {G. Como},
journal = {Annual Reviews in Control},
pages = {80-90},
title = {On resilient control of dynamical flow networks},
volume = {43},
year = {2017},
doi = {10.1016/j.arcontrol.2017.01.001},
}
@article{SC:19,
author = {S. Coogan},
journal = {Automatica},
pages = {349-357},
title = {A contractive approach to separable {Lyapunov}
functions for monotone systems},
volume = {106},
year = {2019},
doi = {10.1016/j.automatica.2019.05.001},
}
@article{YK-BB-MC:20,
author = {Y. Kawano and B. Besselink and M. Cao},
journal = {IEEE Transactions on Automatic Control},
number = {8},
pages = {3486-3501},
title = {Contraction Analysis of Monotone Systems via
Separable Functions},
volume = {65},
year = {2020},
abstract = {In this paper, we study incremental stability of
monotone nonlinear systems through contraction
analysis. We provide sufficient conditions for
incremental asymptotic stability in terms of the Lie
derivatives of differential one-forms or Lie brackets
of vector fields. These conditions can be viewed as
sum- or max-separable conditions, respectively. For
incremental exponential stability, we show that the
existence of such separable functions is both
necessary and sufficient under standard assumptions
for the converse Lyapunov theorem of exponential
stability. As a by-product, we also provide necessary
and sufficient conditions for exponential stability
of positive linear time-varying systems. The results
are illustrated through examples.},
doi = {10.1109/TAC.2019.2944923},
}
@article{SJ-AD-FB:20r,
author = {S. Jafarpour and A. Davydov and F. Bullo},
journal = {IEEE Transactions on Automatic Control},
note = {To appear},
title = {{Non-Euclidean} Contraction Theory for Monotone and
Positive Systems},
year = {2023},
abstract = {In this note we study strong contractivity of
monotone systems and equilibrium-contractivity of
positive systems with respect to non-Euclidean norms.
We first introduce the notion of conic matrix measure
and study its properties. Using conic matrix measures
and weak pairings, we characterize strongly
contracting monotone systems with respect to
non-Euclidean norms. This framework leads to novel
results on (i) the stability of monotone separable
systems, (ii) the strong contractivity of excitatory
Hopfield neural networks, and (iii) a strong version
of the Matrosov-Bellman comparison lemma. We also
characterize equilibrium-contracting positive systems
with respect to non-Euclidean norms and provide a
sufficient condition for equilibrium-contractivity
using conic matrix measures. This framework leads to
novel results on (i) the equilibrium-contractivity of
positive separable systems, and (ii) a
comparison-based framework for interconnected
systems.},
doi = {10.1109/TAC.2022.3224094},
}
@article{AMT:52,
author = {A. M. Turing},
journal = {Philosophical Transactions of the Royal Society of
London. Series B, Biological Sciences},
number = {641},
pages = {37-72},
title = {The chemical basis of morphogenesis},
volume = {237},
year = {1952},
doi = {10.1098/rstb.1952.0012},
}
@article{BCG:65,
author = {B. C. Goodwin},
journal = {Advances in Enzyme Regulation},
pages = {425-437},
title = {Oscillatory behavior in enzymatic control processes},
volume = {3},
year = {1965},
doi = {10.1016/0065-2571(65)90067-1},
}
@article{RFH:61,
author = {R. FitzHugh},
journal = {Biophysical Journal},
number = {6},
pages = {445-466},
title = {Impulses and Physiological States in Theoretical
Models of Nerve Membrane},
volume = {1},
year = {1961},
doi = {10.1016/S0006-3495(61)86902-6},
}
@incollection{MdB-DF-GR-FS:16,
author = {M. {Di~Bernardo} and D. Fiore and G. Russo and
F. Scafuti},
booktitle = {Complex Systems and Networks},
pages = {313--339},
publisher = {Springer},
title = {Convergence, Consensus and Synchronization of Complex
Networks via Contraction Theory},
year = {2016},
abstract = {This chapter reviews several approaches to study
convergence of networks of nonlinear dynamical
systems based on the use of contraction theory.
Rather than studying the properties of the collective
asymptotic solution of interest, the strategy focuses
on finding sufficient conditions for any pair of
trajectories of two agents in the network to converge
towards each other. The key tool is the study, in an
appropriate metric, of the matrix measure of the
agents' or network Jacobian. The effectiveness of the
proposed approach is illustrated via a set of
representative examples.},
doi = {10.1007/978-3-662-47824-0_12},
isbn = {978-3-662-47824-0},
}
@inproceedings{ZA-EDS:14b,
author = {Z. Aminzare and E. D. Sontag},
booktitle = {{IEEE} Conf.\ on Decision and Control},
month = dec,
pages = {3835-3847},
title = {Contraction methods for nonlinear systems: {A} brief
introduction and some open problems},
year = {2014},
doi = {10.1109/CDC.2014.7039986},
}
@article{HT-SJC-JJES:21,
author = {H. Tsukamoto and S.-J. Chung and J.-J. E Slotine},
journal = {Annual Reviews in Control},
pages = {135--169},
title = {Contraction theory for nonlinear stability analysis
and learning-based control: {A} tutorial overview},
volume = {52},
year = {2021},
doi = {10.1016/j.arcontrol.2021.10.001},
}
@article{PS-SH-CK:23,
author = {P. Giesl and S. Hafstein and C. Kawan},
journal = {Journal of Computational Dynamics},
number = {1},
pages = {1-47},
title = {Review on contraction analysis and computation of
contraction metrics},
volume = {10},
year = {2023},
doi = {10.3934/jcd.2022018},
}
@phdthesis{GR:10,
author = {G. Russo},
school = {Universita degli Studi di Napoli Federico II},
title = {Analysis, Control and Synchronization of Nonlinear
Systems and Networks via Contraction Theory: Theory
and Applications},
year = {2010},
}
@phdthesis{ZA:15,
author = {Z. Aminzare},
school = {Rutgers},
title = {On Synchronous Behavior in Complex Nonlinear
Dynamical Systems},
year = {2015},
}
@phdthesis{DW:22,
author = {D. Wu},
school = {Thèse de doctorat de l’université Paris-Saclay et
de Harbin Institute of Technology},
title = {Analyse de contraction des syst{\`e}mes
non-lin{\'e}aires sur des vari{\'e}t{\'e}s
{Riemanniennes}},
year = {2022},
}
@article{JWSP-FB:12za,
author = {J. W. Simpson-Porco and F. Bullo},
journal = {Systems \& Control Letters},
pages = {74-80},
title = {Contraction Theory on {R}iemannian Manifolds},
volume = {65},
year = {2014},
abstract = {Contraction theory is a methodology for assessing the
stability of trajectories of a dynamical system with
respect to one another. In this work, we present the
fundamental results of contraction theory in an
intrinsic, coordinate-free setting, with the
presentation highlighting the underlying geometric
foundation of contraction theory and the resulting
stability properties. We provide coordinate-free
proofs of the main results for autonomous vector
fields, and clarify the assumptions under which the
results hold. We state and prove several interesting
corollaries to the main result, study cascade and
feedback interconnections of contracting systems,
study some simple examples, and highlight how
contraction theory has arisen independently in other
scientific disciplines. We conclude by illustrating
the developed theory for the case of gradient
dynamics.},
doi = {10.1016/j.sysconle.2013.12.016},
}
@article{EA-PAP-JJES:08,
author = {E. M. Aylward and P. A. Parrilo and J-J. E. Slotine},
journal = {Automatica},
number = {8},
pages = {2163-2170},
title = {Stability and robustness analysis of nonlinear
systems via contraction metrics and {SOS}
programming},
volume = {44},
year = {2008},
doi = {10.1016/j.automatica.2007.12.012},
}
@article{QCP-NT-JJES:09,
author = {Q. C. Pham and N. Tabareau and J.-J. E. Slotine},
journal = {IEEE Transactions on Automatic Control},
number = {4},
pages = {816--820},
title = {A contraction theory approach to stochastic
incremental stability},
volume = {54},
year = {2009},
doi = {10.1109/tac.2008.2009619},
}
@article{ZA:22,
author = {Z. Aminzare},
journal = {IEEE Control Systems Letters},
pages = {2311-2316},
title = {Stochastic Logarithmic {Lipschitz} Constants: {A}
Tool to Analyze Contractivity of Stochastic
Differential Equations},
volume = {6},
year = {2022},
abstract = {We introduce the notion of stochastic logarithmic
Lipschitz constants and use these constants to
characterize stochastic contractivity of Itô
stochastic differential equations (SDEs) with
multiplicative noise. We find an upper bound for
stochastic logarithmic Lipschitz constants based on
known logarithmic norms (matrix measures) of the
Jacobian of the drift and diffusion terms of the
SDEs. We discuss noise-induced contractivity in SDEs
and common noise-induced synchronization in network
of SDEs and illustrate the theoretical results on a
noisy Van der Pol oscillator. We show that a
deterministic Van der Pol oscillator is not
contractive, while, adding multiplicative noises
makes the system stochastically contractive.},
doi = {10.1109/LCSYS.2022.3148945},
}
@article{IRM-JJES:14,
author = {I. R. Manchester and J.-J. E. Slotine},
journal = {Systems \& Control Letters},
pages = {32-38},
title = {Transverse contraction criteria for existence,
stability, and robustness of a limit cycle},
volume = {63},
year = {2014},
abstract = {This paper derives a differential contraction
condition for the existence of an orbitally-stable
limit cycle in an autonomous system. This transverse
contraction condition can be represented as a
pointwise linear matrix inequality (LMI), thus
allowing convex optimisation tools such as
sum-of-squares programming to be used to search for
certificates of the existence of a stable limit
cycle. Many desirable properties of contracting
dynamics are extended to this context, including the
preservation of contraction under a broad class of
interconnections. In addition, by introducing the
concepts of differential dissipativity and transverse
differential dissipativity, contraction and
transverse contraction can be established for
interconnected systems via LMI conditions on
component subsystems.},
doi = {10.1016/j.sysconle.2013.10.005},
}
@article{IRM-JJES:18,
author = {Ian R. Manchester and Jean-Jacques E. Slotine},
journal = {{IEEE} Control Systems Letters},
number = {3},
pages = {333--338},
title = {Robust Control Contraction Metrics: {A} Convex
Approach to Nonlinear State-Feedback {$H^\infty$}
Control},
volume = {2},
year = {2018},
abstract = {This letter proposes a new method for robust
state-feedback control design for nonlinear systems.
We introduce robust control contraction metrics
(RCCM), extending the method of control contraction
metrics from stabilization to disturbance attenuation
and robust control. An RCCM is a Riemannian metric
that verifies differential L2-gain bounds in
closed-loop, and guarantees robust stability of
arbitrary trajectories via small gain arguments.
Numerical search for such a metric can be transformed
to a convex optimization problem. We also show that
the associated Riemannian energy can be used as a
robust control Lyapaunov function. A simple
computational example based on jet-engine surge
illustrates the approach.},
doi = {10.1109/lcsys.2018.2836355},
}
@article{BTL-JJES:21,
author = {B. T. Lopez and J.-J. E. Slotine},
journal = {IEEE Control Systems Letters},
number = {1},
pages = {205-210},
title = {Adaptive Nonlinear Control With Contraction Metrics},
volume = {5},
year = {2021},
doi = {10.1109/LCSYS.2020.3000190},
}
@article{HT-SJC:21,
author = {H. Tsukamoto and S.-J. Chung},
journal = {{IEEE} Control Systems Letters},
number = {1},
pages = {211--216},
title = {Neural Contraction Metrics for Robust Estimation and
Control: A Convex Optimization Approach},
volume = {5},
year = {2021},
doi = {10.1109/lcsys.2020.3001646},
}
@article{SS-SMR-VS-JJES-MP:20,
author = {S. Singh and S. M. Richards and V. Sindhwani and
J.-J. E. Slotine and M. Pavone},
journal = {International Journal of Robotics Research},
number = {10-11},
pages = {1123--1150},
title = {Learning stabilizable nonlinear dynamics with
contraction-based regularization},
volume = {40},
year = {2020},
doi = {10.1177/0278364920949931},
}
@article{JSM:90,
author = {J. S. Muldowney},
journal = {Rocky Mountain Journal of Mathematics},
number = {4},
pages = {857-872},
title = {Compound matrices and ordinary differential
equations},
volume = {20},
year = {1990},
doi = {10.1216/rmjm/1181073047},
}
@article{MYL-JSM:96,
author = {M. Y. Li and J. S. Muldowney},
journal = {{SIAM} Journal on Mathematical Analysis},
number = {4},
pages = {1070--1083},
title = {A Geometric Approach to Global-Stability Problems},
volume = {27},
year = {1996},
doi = {10.1137/s0036141094266449},
}
@article{CW-IK-MM:22,
author = {C. Wu and I. Kanevskiy and M. Margaliot},
journal = {Automatica},
pages = {110048},
title = {$k$-contraction: {Theory} and applications},
volume = {136},
year = {2022},
doi = {10.1016/j.automatica.2021.110048},
}
@article{JM-MA:15,
author = {J. Maidens and M. Arcak},
journal = {IEEE Transactions on Automatic Control},
number = {1},
pages = {265-270},
title = {Reachability Analysis of Nonlinear Systems Using
Matrix Measures},
volume = {60},
year = {2015},
abstract = {Matrix measures, also known as logarithmic norms,
have historically been used to provide bounds on the
divergence of trajectories of a system of ordinary
differential equations. In this technical note we use
them to compute guaranteed overapproximations of
reachable sets for nonlinear continuous-time systems
using numerically simulated trajectories and to bound
the accumulation of numerical simulation errors along
simulation traces. Our method employs a user-supplied
bound on the matrix measure of the system's Jacobian
matrix to compute bounds on the behavior of nearby
trajectories, leading to efficient computation of
reachable sets when such bounds are available. We
demonstrate that the proposed technique scales well
to systems with a large number of states.},
doi = {10.1109/TAC.2014.2325635},
}
@article{CF-JK-XJ-SM:18,
author = {C. Fan and J. Kapinski and X. Jin and S. Mitra},
journal = {{ACM} Transactions on Embedded Computing Systems},
number = {1},
pages = {1--28},
title = {Simulation-Driven Reachability Using Matrix Measures},
volume = {17},
year = {2018},
doi = {10.1145/3126685},
}
@article{MM-EDS-TT:16,
author = {M. Margaliot and E. D. Sontag and T. Tuller},
journal = {Automatica},
pages = {178-184},
title = {Contraction after small transients},
volume = {67},
year = {2016},
doi = {10.1016/j.automatica.2016.01.018},
}
@book{FB:22,
author = {F. Bullo},
edition = {{1.6}},
month = jan,
publisher = {Kindle Direct Publishing},
title = {Lectures on Network Systems},
year = {2022},
isbn = {978-1986425643},
url = {http://motion.me.ucsb.edu/book-lns},
}
@book{RG:14,
author = {R. Ghrist},
edition = {1.0},
publisher = {Createspace},
title = {Elementary Applied Topology},
year = {2014},
isbn = {978-1502880857},
}
@article{PCV-SJ-FB:19r,
author = {P. Cisneros-Velarde and S. Jafarpour and F. Bullo},
journal = {IEEE Transactions on Automatic Control},
number = {7},
pages = {3560-3566},
title = {Distributed and Time-Varying Primal-Dual Dynamics via
Contraction Analysis},
volume = {67},
year = {2022},
doi = {10.1109/TAC.2021.3103865},
}
@article{PCV-SJ-FB:20c,
author = {P. Cisneros-Velarde and S. Jafarpour and F. Bullo},
journal = {IEEE Transactions on Automatic Control},
number = {12},
pages = {6710-6715},
title = {Contraction Theory for Dynamical Systems on {Hilbert}
Spaces},
volume = {67},
year = {2022},
abstract = {Contraction theory for dynamical systems on Euclidean
spaces is well-established. For contractive (resp.
semi-contractive) systems, the distance (resp.
semi-distance) between any two trajectories decreases
exponentially fast. For partially contractive
systems, each trajectory converges exponentially fast
to an invariant subspace. In this note, we develop
contraction theory on Hilbert spaces. First, for
time-invariant systems we establish the existence of
a unique globally exponentially stable equilibrium
and provide a novel integral condition for
contractivity. Second, we introduce the notions of
partial and semi-contraction and we provide various
sufficient conditions for time-varying and
time-invariant systems. Finally, we apply the theory
on a reaction-diffusion system.},
doi = {10.1109/TAC.2021.3133270},
}
@inproceedings{SJ-AD-AVP-FB:21f,
author = {S. Jafarpour and A. Davydov and A. V. Proskurnikov and
F. Bullo},
booktitle = {Advances in Neural Information Processing Systems},
month = dec,
title = {Robust Implicit Networks via Non-{Euclidean}
Contractions},
year = {2021},
abstract = {Implicit neural networks, a.k.a., deep equilibrium
networks, are a class of implicit-depth learning
models where function evaluation is performed by
solving a fixed point equation. They generalize
classic feedforward models and are equivalent to
infinite-depth weight-tied feedforward networks.
While implicit models show improved accuracy and
significant reduction in memory consumption, they can
suffer from ill-posedness and convergence
instability. This paper provides a new framework to
design well-posed and robust implicit neural networks
based upon contraction theory for the non-Euclidean
norm $\ell_\infty$. Our framework includes (i) a
novel condition for well-posedness based on one-sided
Lipschitz constants, (ii) an average iteration for
computing fixed-points, and (iii) explicit estimates
on input-output Lipschitz constants. Additionally, we
design a training problem with the well-posedness
condition and the average iteration as constraints
and, to achieve robust models, with the input-output
Lipschitz constant as a regularizer. Our
$\ell_\infty$ well-posedness condition leads to a
larger polytopic training search space than existing
conditions and our average iteration enjoys
accelerated convergence. Finally, we perform several
numerical experiments for function estimation and
digit classification through the MNIST data set. Our
numerical results demonstrate improved accuracy and
robustness of the implicit models with smaller
input-output Lipschitz bounds.},
doi = {10.48550/arXiv.2106.03194},
}
@inproceedings{AD-SJ-AVP-FB:21j,
address = {Canc\'un, M\'exico},
author = {A. Davydov and S. Jafarpour and A. V. Proskurnikov and
F. Bullo},
booktitle = {{IEEE} Conf.\ on Decision and Control},
month = dec,
title = {Non-{Euclidean} Monotone Operator Theory with
Applications to Recurrent Neural Networks},
year = {2022},
doi = {10.1109/CDC51059.2022.9993197},
}
@inproceedings{AD-AVP-FB:21k,
address = {Atlanta, USA},
author = {A. Davydov and A. V. Proskurnikov and F. Bullo},
booktitle = {{A}merican {C}ontrol {C}onference},
month = may,
pages = {1527-1534},
title = {{Non-Euclidean} Contractivity of Recurrent Neural
Networks},
year = {2022},
doi = {10.23919/ACC53348.2022.9867357},
}
@article{KDS-SJ-FB:18f,
author = {K. D. Smith and S. Jafarpour and F. Bullo},
journal = {IEEE Transactions on Automatic Control},
number = {2},
pages = {633-645},
title = {Transient Stability of Droop-Controlled Inverter
Networks with Operating Constraints},
volume = {67},
year = {2022},
abstract = {Due to the rise of distributed energy resources, the
control of networks of grid-forming inverters is now
a pressing issue for power system operation. Droop
control is a popular control strategy in the
literature for frequency control of these inverters.
In this paper, we analyze transient stability in
droop-controlled inverter networks that are subject
to multiple operating constraints. Using two
physically-meaningful Lyapunov-like functions, we
provide two sets of criteria (one mathematical and
one computational) to certify that a post-fault
trajectory achieves frequency synchronization while
respecting operating constraints. We demonstrate two
applications of these results on a modified IEEE RTS
24 test case: estimating the scale of disturbances
with respect to which the system is robust, and
screening for contingencies that threaten transient
stability.},
doi = {10.1109/TAC.2021.3053552},
}
@article{GDP-KDS-FB-MEV:21m,
author = {G. {De~Pasquale} and K. D. Smith and F. Bullo and
M.~E. Valcher},
journal = {IEEE Transactions on Automatic Control},
month = dec,
title = {Dual Seminorms, Ergodic Coefficients, and
Semicontraction Theory},
year = {2022},
doi = {10.48550/arXiv.2201.03103},
}
@article{KDS-FB:22o,
author = {K. D. Smith and F. Bullo},
journal = {IEEE Control Systems Letters},
number = {7},
pages = {919-924},
title = {Contractivity of the Method of Successive
Approximations for Optimal Control},
year = {2023},
abstract = {Strongly contracting dynamical systems have numerous
properties (e.g., incremental ISS), find widespread
applications (e.g., in controls and learning), and
their study is receiving increasing attention. This
letter starts with the simple observation that, given
a strongly contracting system, its adjoint dynamical
system is also strongly contracting, with the same
rate, with respect to the dual norm, under time
reversal. As main implication of this dual
contractivity, we show that the classic Method of
Successive Approximations (MSA), an indirect method
in optimal control, is a contraction mapping for
short optimization intervals or large contraction
rates. Consequently, we establish new convergence
conditions for the MSA algorithm, which further imply
uniqueness of the optimal control and sufficiency of
Pontryagin’s minimum principle under additional
assumptions.},
doi = {10.1109/LCSYS.2022.3228723},
}
@article{VC-FB-GR:22k,
author = {V. Centorrino and F. Bullo and G. Russo},
journal = {Automatica},
month = jul,
note = {Submitted},
title = {Modelling and Contractivity of Neural-Synaptic
Networks with {Hebbian} Learning},
year = {2022},
doi = {10.48550/arXiv.2204.05382},
}
@article{RO-FB-MM:22h,
author = {R. Ofir and F. Bullo and M. Margaliot},
journal = {IEEE Control Systems Letters},
pages = {2731-2736},
title = {Minimum effort decentralized control design for
contracting network systems},
volume = {6},
year = {2022},
abstract = {We consider the problem of making a networked system
contracting by designing ``minimal effort'' local
controllers. Our method combines a hierarchical
contraction characterization and a matrix-balancing
approach to stabilizing a Metzler matrix via minimal
diagonal perturbations. We demonstrate our approach
by designing local controllers that render
contractive a network of FitzHugh–Nagumo neurons
with a general topology of interactions.},
doi = {10.1109/LCSYS.2022.3176196},
}
@article{GR-MDB:09,
author = {G. Russo and M. {Di~Bernardo}},
journal = {IEEE Transactions on Circuits and Systems II: Express
Briefs},
number = {2},
pages = {177--181},
title = {Contraction Theory and Master Stability Function:
Linking Two Approaches to Study Synchronization of
Complex Networks},
volume = {56},
year = {2009},
doi = {10.1109/TCSII.2008.2011611},
}
@article{GR-MDI-JJES:11,
author = {G. Russo and M. {Di~Bernardo} and J.-J. E. Slotine},
journal = {IEEE Transactions on Circuits and Systems~I},
number = {2},
pages = {336--348},
title = {A Graphical Approach to Prove Contraction of
Nonlinear Circuits and Systems},
volume = {58},
year = {2011},
doi = {10.1109/TCSI.2010.2071810},
}
@article{PD-MdB-GR:11,
author = {P. DeLellis and M. {Di~Bernardo} and G. Russo},
journal = {IEEE Transactions on Circuits and Systems I: Regular
Papers},
number = {3},
pages = {576-583},
title = {On {QUAD}, {Lipschitz}, and Contracting Vector Fields
for Consensus and Synchronization of Networks},
volume = {58},
year = {2011},
doi = {10.1109/TCSI.2010.2072270},
}
@article{MDB-DL-GR:14,
author = {M. {Di~Bernardo} and D. Liuzza and G. Russo},
journal = {SIAM Journal on Control and Optimization},
number = {5},
pages = {3203-3227},
title = {Contraction Analysis for a Class of NonDifferentiable
Systems with Applications to Stability and Network
Synchronization},
volume = {52},
year = {2014},
doi = {10.1137/120883001},
}
@article{DF-SJH-MDB:16,
author = {D. Fiore and S. J. Hogan and M. {Di~Bernardo}},
journal = {Automatica},
pages = {279-288},
title = {Contraction analysis of switched systems via
regularization},
volume = {73},
year = {2016},
abstract = {We study incremental stability and convergence of
switched (bimodal) Filippov systems via contraction
analysis. In particular, by using results on
regularization of switched dynamical systems, we
derive sufficient conditions for convergence of any
two trajectories of the Filippov system between each
other within some region of interest. We then apply
these conditions to the study of different classes of
Filippov systems including piecewise smooth (PWS)
systems, piecewise affine (PWA) systems and relay
feedback systems. We show that contrary to previous
approaches, our conditions allow the system to be
studied in metrics other than the Euclidean norm. The
theoretical results are illustrated by numerical
simulations on a set of representative examples that
confirm their effectiveness and ease of application.},
doi = {10.1016/j.automatica.2016.06.028},
}
@article{WL-MdB:16,
author = {W. Lu and M. {Di~Bernardo}},
journal = {Automatica},
pages = {1-8},
title = {Contraction and incremental stability of switched
{Carathéodory} systems using multiple norms},
volume = {70},
year = {2016},
doi = {10.1016/j.automatica.2016.02.039},
}
@article{MM-EDS-TT:14,
author = {M. Margaliot and E. D. Sontag and T. Tuller},
journal = {PLoS One},
number = {5},
pages = {e96039},
title = {Entrainment to Periodic Initiation and Transition
Rates in a Computational Model for Gene Translation},
volume = {9},
year = {2014},
doi = {10.1371/journal.pone.0096039},
}
@article{SC-MM:19,
author = {S. Coogan and M. Margaliot},
journal = {IEEE Transactions on Automatic Control},
number = {2},
pages = {847-853},
title = {Approximating the Steady-State Periodic Solutions of
Contractive Systems},
volume = {64},
year = {2019},
abstract = {We consider contractive systems whose trajectories
evolve on a compact and convex state-space. It is
well-known that if the time-varying vector field of
the system is periodic, then the system admits a
unique globally asymptotically stable periodic
solution. Obtaining explicit information on this
periodic solution and its dependence on various
parameters is important both theoretically and in
numerous applications. We develop an approach for
approximating such a periodic trajectory using the
periodic trajectory of a simpler system (e.g., an LTI
system). The approximation includes an error bound
that is based on the input-to-state stability
property of contractive systems. We show that in some
cases, this error bound can be computed explicitly.
We also use the bound to derive a new theoretical
result, namely, that a contractive system with an
additive periodic input behaves like a low-pass
filter. We demonstrate our results using several
examples from systems biology.},
doi = {10.1109/TAC.2018.2838054},
}
@article{JM-GR-RS:19,
author = {J. Monteil and G. Russo and R. Shorten},
journal = {Automatica},
pages = {198-205},
title = {On $\mathcal{L}_{\infty}$ string stability of
nonlinear bidirectional asymmetric heterogeneous
platoon systems},
volume = {105},
year = {2019},
abstract = {This paper is concerned with the study of
bidirectionally coupled platoon systems. The case
considered is when the vehicles are heterogeneous and
the coupling can be nonlinear and asymmetric. For
such systems, a sufficient condition for L∞ string
stability is presented. The effectiveness of our
approach is illustrated via a numerical example,
where it is shown how our result can be recast as an
optimization problem, allowing to design the control
protocol for each vehicle independently on the other
vehicles and hence leading to a bottom-up approach
for the design of string stable systems able to track
a time-varying reference speed.},
doi = {10.1016/j.automatica.2019.03.025},
}
@article{SX-GR-RHM:21,
author = {S. Xie and G. Russo and R. H. Middleton},
journal = {IEEE Transactions on Control of Network Systems},
number = {3},
pages = {1128-1138},
title = {Scalability in Nonlinear Network Systems Affected by
Delays and Disturbances},
volume = {8},
year = {2021},
doi = {10.1109/TCNS.2021.3058934},
}
@article{ZA-EDS:13,
author = {Z. Aminzare and E. D. Sontag},
journal = {Nonlinear Analysis: Theory, Methods \& Applications},
pages = {31-49},
title = {Logarithmic {Lipschitz} norms and diffusion-induced
instability},
volume = {83},
year = {2013},
doi = {10.1016/j.na.2013.01.001},
}
@incollection{ZA-YS-MA-EDS:14,
author = {Z. Aminzare and Y. Shafi and M. Arcak and
E. D. Sontag},
booktitle = {A Systems Theoretic Approach to Systems and Synthetic
Biology I: Models and System Characterizations},
chapter = {3},
pages = {73-101},
publisher = {Springer},
title = {Guaranteeing Spatial Uniformity in Reaction-Diffusion
Systems Using Weighted {$L_2$} Norm Contractions},
year = {2014},
doi = {10.1007/978-94-017-9041-3_3},
isbn = {9789401790413},
}
@article{ZA-BD-END-NEL:18,
author = {Z. Aminzare and B. Dey and E. N. Davison and
N. E. Leonard},
journal = {Journal of Nonlinear Science},
title = {Cluster Synchronization of Diffusively Coupled
Nonlinear Systems: {A} Contraction-Based Approach},
year = {2018},
abstract = {Finding the conditions that foster synchronization in
networked nonlinear systems is critical to
understanding a wide range of biological and
mechanical systems. However, the conditions proved in
the literature for synchronization in nonlinear
systems with linear coupling, such as has been used
to model neuronal networks, are in general not strict
enough to accurately determine the system behavior.
We leverage contraction theory to derive new
sufficient conditions for cluster synchronization in
terms of the network structure, for a network where
the intrinsic nonlinear dynamics of each node may
differ. Our result requires that network connections
satisfy a cluster-input-equivalence condition, and we
explore the influence of this requirement on network
dynamics. For application to networks of nodes with
FitzHugh--Nagumo dynamics, we show that our new
sufficient condition is tighter than those found in
previous analyses that used smooth or nonsmooth
Lyapunov functions. Improving the analytical
conditions for when cluster synchronization will
occur based on network configuration is a significant
step toward facilitating understanding and control of
complex networked systems.},
doi = {10.1007/s00332-018-9457-y},
}
@article{KS-SJC-JJES:10,
author = {K. Seo and S. J. Chung and J.-J. E. Slotine},
journal = {Autonomous Robots},
pages = {247-269},
title = {{CPG}-based control of a turtle-like underwater
vehicle},
volume = {28},
year = {2010},
abstract = {This paper presents biologically inspired control
strategies for an autonomous underwater vehicle (AUV)
propelled by flapping fins that resemble the
paddle-like forelimbs of a sea turtle. Our proposed
framework exploits limit cycle oscillators and
diffusive couplings, thereby constructing coupled
nonlinear oscillators, similar to the central pattern
generators (CPGs) in animal spinal cords. This paper
first presents rigorous stability analyses and
experimental results of CPG-based control methods
with and without actuator feedback to the CPG. In
these methods, the CPG module generates synchronized
oscillation patterns, which are sent to
position-servoed flapping fin actuators as a
reference input. In order to overcome the limitation
of the open-loop CPG that the synchronization is
occurring only between the reference signals, this
paper introduces a new single-layered CPG method,
where the CPG and the physical layers are combined as
a single layer, to ensure the synchronization of the
physical actuators in the presence of external
disturbances. The key idea is to replace nonlinear
oscillators in the conventional CPG models with
physical actuators that oscillate due to nonlinear
state feedback of the actuator states. Using
contraction theory, a relatively new nonlinear
stability tool, we show that coupled nonlinear
oscillators globally synchronize to a specific
pattern that can be stereotyped by an outer-loop
controller. Results of experimentation with a
turtle-like AUV show the feasibility of the proposed
control laws.},
doi = {10.1007/s10514-009-9169-0},
}
@incollection{EDS:10,
author = {E. D. Sontag},
booktitle = {Perspectives in Mathematical System Theory, Control,
and Signal Processing},
editor = {J. C. Willems and S. Hara and Y. Ohta and H. Fujioka},
pages = {217-228},
publisher = {Springer},
title = {Contractive Systems with Inputs},
year = {2010},
isbn = {978-3540939177},
}
@inproceedings{AH-ES-DDV:15,
author = {A. Hamadeh and E. Sontag and D. {Del~Vecchio}},
booktitle = {{IEEE} Conf.\ on Decision and Control},
month = dec,
pages = {7689-7694},
title = {A contraction approach to input tracking via high
gain feedback},
year = {2015},
abstract = {This paper adopts a contraction approach to study
exogenous input tracking in dynamical systems under
high gain proportional output feedback. We give
conditions under which contraction of a nonlinear
system's tracking error implies input to output
stability from the input signal's time derivatives to
the tracking error. This result is then used to
demonstrate that the negative feedback connection of
plants composed of two strictly positive real
subsystems in cascade can follow external inputs with
tracking errors that can be made arbitrarily small by
applying a sufficiently large feedback gain. We
utilize this result to design a biomolecular feedback
regulation scheme for a synthetic genetic sensor
model, making it robust to variations in the
availability of a cellular resource required for
protein production.},
doi = {10.1109/CDC.2015.7403435},
}
@article{HDN-TLV-KT-JJES:18,
author = {H. D. {Nguyen} and T. L. {Vu} and K. {Turitsyn} and
J.-J. E. {Slotine}},
journal = {IEEE Control Systems Letters},
number = {4},
pages = {755-760},
title = {Contraction and Robustness of Continuous Time
Primal-Dual Dynamics},
volume = {2},
year = {2018},
doi = {10.1109/LCSYS.2018.2847408},
}
@article{MTA-JJES:17,
author = {M. T. Angulo and J.-J. E. Slotine},
journal = {{IEEE} Transactions on Automatic Control},
number = {8},
pages = {4080--4085},
title = {Qualitative Stability of Nonlinear Networked Systems},
volume = {62},
year = {2017},
doi = {10.1109/tac.2016.2617780},
}
@article{MTA-YYL-JJES:15,
author = {M. T. Angulo and Y.-Y. Liu and J.-J. E. Slotine},
journal = {Nature Physics},
number = {10},
pages = {848--852},
title = {Network motifs emerge from interconnections that
favour stability},
volume = {11},
year = {2015},
doi = {10.1038/nphys3402},
}
@article{SB-JJES:15,
author = {S. Bonnabel and J.-J. E. Slotine},
journal = {{IEEE} Transactions on Automatic Control},
number = {2},
pages = {565--569},
title = {A Contraction Theory-Based Analysis of the Stability
of the Deterministic Extended {Kalman} Filter},
volume = {60},
year = {2015},
doi = {10.1109/tac.2014.2336991},
}
@article{PMW-JJES:20,
author = {P. M. Wensing and J.-J. E. Slotine},
journal = {PLoS One},
number = {8},
pages = {1-29},
title = {Beyond convexity --- {Contraction} and global
convergence of gradient descent},
volume = {15},
year = {2020},
doi = {10.1371/journal.pone.0236661},
}
@article{LK-ML-JJES-EKM:20,
author = {L. Kozachkov and M. Lundqvist and J.-J. E. Slotine and
E. K. Miller},
journal = {PLoS Computational Biology},
number = {8},
pages = {1-15},
title = {Achieving stable dynamics in neural circuits},
volume = {16},
year = {2020},
abstract = {The brain consists of many interconnected networks
with time-varying, partially autonomous activity.
There are multiple sources of noise and variation yet
activity has to eventually converge to a stable,
reproducible state (or sequence of states) for its
computations to make sense. We approached this
problem from a control-theory perspective by applying
contraction analysis to recurrent neural networks.
This allowed us to find mechanisms for achieving
stability in multiple connected networks with
biologically realistic dynamics, including synaptic
plasticity and time-varying inputs. These mechanisms
included inhibitory Hebbian plasticity, excitatory
anti-Hebbian plasticity, synaptic sparsity and
excitatory-inhibitory balance. Our findings shed
light on how stable computations might be achieved
despite biological complexity. Crucially, our
analysis is not limited to analyzing the stability of
fixed geometric objects in state space (e.g points,
lines, planes), but rather the stability of state
trajectories which may be complex and time-varying.},
doi = {10.1371/journal.pcbi.1007659},
}
@article{JJ-TIF:10,
author = {Jouffroy, J. and Fossen, T. I.},
journal = {Modeling, Identification and Control},
number = {3},
pages = {93--106},
title = {A Tutorial on Incremental Stability Analysis using
Contraction Theory},
volume = {31},
year = {2010},
doi = {10.4173/mic.2010.3.2},
}
@inproceedings{JJ-JJES:04,
author = {J. {Jouffroy} and J.-J. E. {Slotine}},
booktitle = {{IEEE} Conf.\ on Decision and Control},
pages = {2537-2543},
title = {Methodological remarks on contraction theory},
volume = {3},
year = {2004},
doi = {10.1109/CDC.2004.1428824},
}
@article{RC:1930,
author = {Caccioppoli, Renato},
journal = {Rendiconti dell'Accademia Nazionale dei Lincei},
pages = {794--799},
title = {Un teorema generale sull'esistenza di elementi uniti
in una trasformazione funzionale},
volume = {11},
year = {1930},
}
@book{RPA-MM-DOR:01,
author = {R. P. Agarwal and M. Meehan and D. {O'Regan}},
publisher = {Cambridge University Press},
title = {Fixed Point Theory and Applications},
year = {2001},
isbn = {0-521-80250-4},
}
@book{DPB-JNT:97,
author = {D. P. Bertsekas and J. N. Tsitsiklis},
publisher = {Athena Scientific},
title = {Parallel and Distributed Computation: Numerical
Methods},
year = {1997},
annote = {This book is a comprehensive and theoretically sound
treatment of parallel and distributed numerical
methods. It focuses on algorithms that are naturally
suited for massive parallelization, and it explores
the fundamental convergence, rate of convergence,
communication, and synchronization issues associated
with such algorithms.},
isbn = {1886529019},
}
@article{TCL:85,
author = {T. C. Lim},
journal = {Journal of Mathematical Analysis and Applications},
number = {2},
pages = {436--441},
title = {On fixed point stability for set-valued contractive
mappings with applications to generalized
differential equations},
volume = {110},
year = {1985},
doi = {10.1016/0022-247X(85)90306-3},
}
@article{GB:57,
author = {G. Birkhoff},
journal = {Transactions of the American Mathematical Society},
number = {1},
pages = {219--227},
title = {Extensions of {J}entzsch's theorem},
volume = {85},
year = {1957},
doi = {10.2307/1992971},
}
@article{PJB:73,
author = {P. J. Bushell},
journal = {Archive for Rational Mechanics and Analysis},
number = {4},
pages = {330--338},
title = {Hilbert's metric and positive contraction mappings in
a {Banach} space},
volume = {52},
year = {1973},
doi = {10.1007/BF00247467},
}
@article{EK-JWP:82,
author = {E. Kohlberg and J. W. Pratt},
journal = {Mathematics of Operations Research},
number = {2},
pages = {198--210},
title = {The contraction mapping approach to the
Perron-Frobenius theory: Why {Hilbert's} metric?},
volume = {7},
year = {1982},
doi = {10.1287/moor.7.2.198},
}
@article{UK:86,
author = {U. Krause},
journal = {Journal of Mathematical Economics},
number = {3},
pages = {275--282},
title = {Perron's stability theorem for non-linear mappings},
volume = {15},
year = {1986},
doi = {10.1016/0304-4068(86)90016-9},
}
@article{UK:94,
author = {U. Krause},
journal = {Journal of Mathematical Analysis and Applications},
number = {1},
pages = {182--202},
title = {Relative stability for ascending and positively
homogeneous operators on {Banach} spaces},
volume = {188},
year = {1994},
doi = {10.1006/jmaa.1994.1420},
}
@article{UK:01,
author = {U. Krause},
journal = {Nonlinear Analysis, Theory, Methods \& Applications},
number = {3},
pages = {1457-1466},
title = {Concave {Perron-Frobenius Theory} and applications},
volume = {47},
year = {2001},
doi = {10.1016/S0362-546X(01)00281-4},
}
@book{BL-RN:12,
author = {Lemmens, B. and Nussbaum, R.},
publisher = {Cambridge University Press},
series = {Cambridge Tracts in Mathematics},
title = {Nonlinear {P}erron-{F}robenius Theory},
year = {2012},
isbn = {9780521898812},
}
@article{ACR:63,
author = {A. C. Thompson},
journal = {Proceedings of the American Mathematical Society},
month = jun,
number = {3},
pages = {438},
title = {On Certain Contraction Mappings in a Partially
Ordered Vector Space},
volume = {14},
year = {1963},
doi = {10.2307/2033816},
}
@article{PB:93,
author = {P. Bougerol},
journal = {SIAM Journal on Control and Optimization},
month = jul,
number = {4},
pages = {942--959},
title = {Kalman Filtering with Random Coefficients and
Contractions},
volume = {31},
year = {1993},
doi = {10.1137/0331041},
}
@article{RajB:03,
author = {R. Bhatia},
journal = {Linear Algebra and its Applications},
month = dec,
pages = {211--220},
title = {On the exponential metric increasing property},
volume = {375},
year = {2003},
doi = {10.1016/s0024-3795(03)00647-5},
}
@article{JL-YL:07,
author = {Jimmie Lawson and Yongdo Lim},
journal = {SIAM Journal on Control and Optimization},
number = {3},
pages = {930--951},
title = {A {Birkhoff} Contraction Formula with Applications to
{Riccati} Equations},
volume = {46},
year = {2007},
doi = {10.1137/050637637},
}
@article{HL-YL:08,
author = {H. Lee and Y. Lim},
journal = {Nonlinearity},
number = {4},
pages = {857--878},
title = {Invariant metrics, contractions and nonlinear matrix
equations},
volume = {21},
year = {2008},
doi = {10.1088/0951-7715/21/4/011},
}
@article{SML:58,
author = {S. M. Lozinskii},
journal = {Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika},
note = {(in Russian)},
pages = {52--90},
publisher = {Kazan (Volga region) Federal University},
title = {Error estimate for numerical integration of ordinary
differential equations. {I}},
volume = {5},
year = {1958},
url = {http://mi.mathnet.ru/eng/ivm2980},
}
@phdthesis{GD:58,
author = {G. Dahlquist},
school = {(Reprinted in Trans.\ Royal Inst.\ of Technology,
No.\ 130, Stockholm, Sweden, 1959)},
title = {Stability and error bounds in the numerical
integration of ordinary differential equations},
year = {1958},
}
@book{WAC:1965,
author = {W. A. Coppel},
publisher = {Heath},
title = {Stability and Asymptotic Behavior of Differential
Equations},
year = {1965},
isbn = {0669190187},
}
@book{SB-LV:04,
author = {S. Boyd and L. Vandenberghe},
publisher = {Cambridge University Press},
title = {Convex Optimization},
year = {2004},
isbn = {0521833787},
}
@misc{MG-SB:11-cvx,
author = {M. Grant and S. Boyd},
month = mar,
title = {{CVX}: Matlab Software for Disciplined Convex
Programming, version 2.1},
year = {2014},
url = {http://cvxr.com/cvx},
}
@book{SB-LEG-EF-VB:94,
author = {S. Boyd and L. {El~Ghaoui} and E. Feron and
V. Balakrishnan},
publisher = {SIAM},
title = {Linear Matrix Inequalities in System and Control
Theory},
year = {1994},
isbn = {089871334X},
}
@article{OP-MV:06,
author = {O. Pastravanu and M. Voicu},
journal = {Linear Algebra and its Applications},
number = {2},
pages = {299-310},
title = {Generalized matrix diagonal stability and linear
dynamical systems},
volume = {419},
year = {2006},
abstract = {Let A=(aij) be a real square matrix and p in 1,
infinity. We present two analogous developments. One
for Schur stability and the discrete-time dynamical
system x(t+1)=Ax(t), and the other for Hurwitz
stability and the continuous-time dynamical system
x'(t)=Ax(t). Here is a description of the latter
development. For A, we define and study Hurwitz
diagonal stability with respect to p-norms,
abbreviated HDSp. HDS2 is the usual concept of
diagonal stability. A is HDSp implies Re lambda <0
for every eigenvalue, which means A is Hurwitz
stable, abbreviated HS. When the off-diagonal
elements of A are nonnegative, A is HS iff A is HDSp
for all p. For the dynamical system x'(t)=Ax(t), we
define diagonally invariant exponential stability
relative to the p-norm, abbreviated DIESp, meaning
there exist time-dependent sets, which decrease
exponentially and are invariant with respect to the
system. We show that DIESp is a special type of
exponential stability and the dynamical system has
this property iff A is HDSp.},
doi = {10.1016/j.laa.2006.04.021},
}
@article{PJM:77,
author = {P. J. Moylan},
journal = {Linear Algebra and its Applications},
number = {1},
pages = {53-58},
title = {Matrices with positive principal minors},
volume = {17},
year = {1977},
abstract = {A new necessary and sufficient condition is given for
all principal minors of a square matrix to be
positive. A special subclass of such matrices, called
quasidominant matrices, is also examined.},
doi = {10.1016/0024-3795(77)90040-4},
}
@book{RAH-CRJ:94,
author = {R. A. Horn and C. R. Johnson},
publisher = {Cambridge University Press},
title = {Topics in Matrix Analysis},
year = {1994},
isbn = {0521467136},
}
@book{RAH-CRJ:12,
author = {R. A. Horn and C. R. Johnson},
edition = {2nd},
publisher = {Cambridge University Press},
title = {Matrix Analysis},
year = {2012},
isbn = {0521548233},
}
@article{AAA-EdK-GH:19,
author = {A. A. Ahmadi and E. {de~Klerk} and G. Hall},
journal = {SIAM Journal on Optimization},
number = {1},
pages = {399-422},
title = {Polynomial Norms},
volume = {29},
year = {2019},
doi = {10.1137/18M1172843},
}
@article{AP:18,
author = {A. Polyakov},
journal = {International Journal of Robust and Nonlinear
Control},
number = {3},
pages = {682--701},
title = {Sliding mode control design using canonical
homogeneous norm},
volume = {29},
year = {2018},
doi = {10.1002/rnc.4058},
}
@article{PL-HKF:72,
author = {P. Lancaster and H. K. Farahat},
journal = {Mathematics of Computation},
number = {118},
pages = {401-414},
publisher = {American Mathematical Society},
title = {Norms on Direct Sums and Tensor Products},
volume = {26},
year = {1972},
doi = {10.2307/2005167},
}
@book{CAD-MV:1975,
author = {C. A. Desoer and M. Vidyasagar},
publisher = {Academic Press},
title = {Feedback Systems: Input-Output Properties},
year = {1975},
doi = {10.1137/1.9780898719055},
isbn = {978-0-12-212050-3},
}
@article{JS-CW:62,
author = {Stoer, J. and Witzgall, C.},
journal = {Numerische Mathematik},
pages = {158-171},
title = {Transformations by diagonal matrices in a normed
space},
volume = {4},
year = {1962},
doi = {10.1007/BF01386309},
}
@article{JA:96,
author = {J. Albrecht},
journal = {Linear Algebra and its Applications},
number = {1},
pages = {255-258},
title = {Minimal norms of nonnegative irreducible matrices},
volume = {249},
year = {1996},
doi = {10.1016/0024-3795(95)00360-6},
}
@article{WB:81,
author = {W. Bunse},
journal = {SIAM Journal on Numerical Analysis},
number = {4},
pages = {693-704},
title = {A Class of Diagonal Transformation Methods for the
Computation of the Spectral Radius of a Nonnegative
Irreducible Matrix},
volume = {18},
year = {1981},
doi = {10.1137/0718046},
}
@article{PVA:91,
author = {P. {Van~At}},
journal = {Linear Algebra and its Applications},
pages = {93-123},
title = {Diagonal transformation methods for computing the
maximal eigenvalue and eigenvector of a nonnegative
irreducible matrix},
volume = {148},
year = {1991},
abstract = {Sufficient conditions for the convergence of diagonal
transformation methods for computing the maximal
eigenvalue and eigenvector of nonnegative irreducible
matrices are formulated in a general form. Using
these new sufficient conditions, we can easily
consider the convergence of known methods and
construct many new ones.},
doi = {10.1016/0024-3795(91)90089-F},
}
@article{ICFI-TMS:11,
author = {I. C. F. Ipsen and T. M. Selee},
journal = {SIAM Journal on Matrix Analysis and Applications},
number = {1},
pages = {153-200},
title = {Ergodicity Coefficients Defined by Vector Norms},
volume = {32},
year = {2011},
doi = {10.1137/090752948},
}
@article{YN-AN:15,
author = {Y. Nesterov and A. Nemirovski},
journal = {Applied Mathematics and Computation},
pages = {58-65},
title = {Finding the stationary states of {Markov} chains by
iterative methods},
volume = {255},
year = {2015},
abstract = {In this paper, we develop new methods for
approximating dominant eigenvector of
column-stochastic matrices. We analyze the Google
matrix, and present an averaging scheme with linear
rate of convergence in terms of 1-norm distance. For
extending this convergence result onto general case,
we assume existence of a positive row in the matrix.
Our new numerical scheme, the Reduced Power Method
(RPM), can be seen as a proper averaging of the power
iterates of a reduced stochastic matrix. We analyze
also the usual Power Method (PM) and obtain
convenient conditions for its linear rate of
convergence with respect to 1-norm.},
doi = {10.1016/j.amc.2014.04.053},
}
@article{GS:06,
author = {G. S{\"o}derlind},
journal = {BIT Numerical Mathematics},
number = {3},
pages = {631--652},
title = {The logarithmic norm. {History} and modern theory},
volume = {46},
year = {2006},
abstract = {In his 1958 thesis Stability and Error Bounds,
Germund Dahlquist introduced the logarithmic norm in
order to derive error bounds in initial value
problems, using differential inequalities that
distinguished between forward and reverse time
integration. Originally defined for matrices, the
logarithmic norm can be extended to bounded linear
operators, but the extensions to nonlinear maps and
unbounded operators have required a functional
analytic redefinition of the concept. This compact
survey is intended as an elementary, but broad and
largely self-contained, introduction to the versatile
and powerful modern theory. Its wealth of
applications range from the stability theory of IVPs
and BVPs, to the solvability of algebraic, nonlinear,
operator, and functional equations.},
doi = {10.1007/s10543-006-0069-9},
}
@incollection{GD:1976,
author = {Dahlquist, Germund},
booktitle = {Numerical Analysis},
editor = {Watson, G. A.},
pages = {60--72},
publisher = {Springer},
title = {Error analysis for a class of methods for stiff
non-linear initial value problems},
year = {1976},
doi = {10.1007/BFb0080115},
}
@article{TS:75,
author = {T. Str{\"o}m},
journal = {SIAM Journal on Numerical Analysis},
number = {5},
pages = {741--753},
title = {On logarithmic norms},
volume = {12},
year = {1975},
doi = {10.1137/0712055},
}
@book{MV:78-book,
author = {M. Vidyasagar},
publisher = {Prentice Hall},
title = {Nonlinear Systems Analysis},
year = {1978},
doi = {10.1137/1.9780898719185},
isbn = {0136232809},
}
@book{KD-JGV:84,
author = {Dekker, K. and Verwer, J. G.},
publisher = {North-Holland},
title = {Stability of Runge-Kutta Methods for Stiff Nonlinear
Differential Equations},
year = {1984},
isbn = {0-444-87634-0},
}
@book{EH-SPN-GW:93,
author = {E. Hairer and S. P. N\o{}rsett and G. Wanner},
publisher = {Springer},
title = {Solving Ordinary Differential Equations I. Nonstiff
Problems},
year = {1993},
doi = {10.1007/978-3-540-78862-1},
}
@book{DDS:78,
author = {D. D. {\v S}iljak},
publisher = {North-Holland},
title = {Large-Scale Dynamic Systems Stability \& Structure},
year = {1978},
isbn = {0486462854},
}
@techreport{ADL:10,
author = {A. D. Lewis},
institution = {Queen’s University, Kingston, Ontario, Canada},
title = {A top nine list: {M}ost popular induced matrix norms},
year = {2010},
url = {https://mast.queensu.ca/~andrew/notes/pdf/2010a.pdf},
}
@article{IH-BGC:99,
author = {I. Higueras and B. Garcia-Celayeta},
journal = {{SIAM} Journal on Matrix Analysis and Applications},
number = {3},
pages = {646--666},
title = {Logarithmic Norms for Matrix Pencils},
volume = {20},
year = {1999},
doi = {10.1137/s0895479897325955},
}
@article{YF-KAP-XF:93,
author = {Y. Fang and K. A. Loparo and X. Feng},
journal = {International Journal of Control},
number = {4},
pages = {969-977},
title = {Sufficient conditions for the stability of interval
matrices},
volume = {58},
year = {1993},
abstract = {The stability of interval dynamical systems is
studied. Sufficient conditions for the polytope of
interval matrices are examined and some of the proofs
are greatly simplified. More generally sufficient
conditions are obtained and the approach taken in the
new proofs has the potential for further
generalizations of the result obtained in this
paper.},
doi = {10.1080/00207179308923038},
}
@article{HQ-JP-ZBX:01,
author = {H. Qiao and J. Peng and Z.-B. Xu},
journal = {IEEE Transactions on Neural Networks},
number = {2},
pages = {360-370},
title = {Nonlinear measures: {A} new approach to exponential
stability analysis for {Hopfield}-type neural
networks},
volume = {12},
year = {2001},
abstract = {In this paper, a new concept called nonlinear measure
is introduced to quantify stability of nonlinear
systems in the way similar to the matrix measure for
stability of linear systems. Based on the new
concept, a novel approach for stability analysis of
neural networks is developed. With this approach, a
series of new sufficient conditions for global and
local exponential stability of Hopfield type neural
networks is presented, which generalizes those
existing results. By means of the introduced
nonlinear measure, the exponential convergence rate
of the neural networks to stable equilibrium point is
estimated, and, for local stability, the attraction
region of the stable equilibrium point is
characterized. The developed approach can be
generalized to stability analysis of other general
nonlinear systems.},
doi = {10.1109/72.914530},
}
@article{WH-JC:09,
author = {W. He and J. Cao},
journal = {Nonlinear Dynamics},
pages = {55-65},
title = {Exponential synchronization of chaotic neural
networks: a matrix measure approach},
volume = {55},
year = {2009},
doi = {10.1007/s11071-008-9344-4},
}
@inproceedings{FB-PCV-AD-SJ:21e,
author = {F. Bullo and P. Cisneros-Velarde and A. Davydov and
S. Jafarpour},
booktitle = {{IEEE} Conf.\ on Decision and Control},
month = dec,
title = {From Contraction Theory to Fixed Point Algorithms on
{Riemannian} and non-{Euclidean} Spaces},
year = {2021},
doi = {10.1109/CDC45484.2021.9682883},
}
@article{OP-MHM:10,
author = {O. Pastravanu and M. H. Matcovschi},
journal = {Journal of the Franklin Institute},
number = {3},
pages = {627-640},
title = {Linear time-variant systems: {Lyapunov} functions and
invariant sets defined by {H\"older} norms},
volume = {347},
year = {2010},
abstract = {For linear time-variant systems x˙(t)=A(t)x(t), we
consider Lyapunov function candidates of the form
Vp(x,t)=||H(t)x||p, with 1≤p≤∞, defined by
continuously differentiable and non-singular
matrix-valued functions, H(t):R+→Rn×n. We prove
that the traditional framework based on quadratic
Lyapunov functions represents a particular case (i.e.
p=2) of a more general scenario operating in similar
terms for all Hölder p-norms. We propose a unified
theory connecting, by necessary and sufficient
conditions, the properties of (i) the matrix-valued
function H(t), (ii) the Lyapunov function candidate
Vp(x,t) and (iii) the time-dependent set
Xp(t)={x∈Rn|||H(t)x||p≤e−rt}, with r≥0. This
theory allows the construction of four distinct types
of Lyapunov functions and, equivalently, four
distinct types of sets which are invariant with
respect to the system trajectories. Subsequently, we
also get criteria for testing stability, uniform
stability, asymptotic stability and exponential
stability. For all types of Lyapunov functions, the
matrix-valued function H(t) is a solution to a matrix
differential inequality (or, equivalently, matrix
differential equation) expressed in terms of matrix
measures corresponding to Hölder p-norms. Such an
inequality (or equation) generalizes the role played
by the Lyapunov inequality (equation) in the
classical case when p=2. Finally, we discuss the
diagonal-type Lyapunov functions that are easier to
handle (including the generalized Lyapunov
inequality) because of the diagonal form of H(t).},
doi = {10.1016/j.jfranklin.2010.01.002},
}
@article{RV:20,
author = {R. Vrabel},
journal = {{IEEE} Transactions on Automatic Control},
number = {4},
pages = {1647--1651},
title = {A Note on Uniform Exponential Stability of Linear
Periodic Time-Varying Systems},
volume = {65},
year = {2020},
doi = {10.1109/tac.2019.2927949},
}
@article{ED:75,
author = {E. Deutsch},
journal = {Numerische Mathematik},
number = {1},
pages = {49--51},
title = {On matrix norms and logarithmic norms},
volume = {24},
year = {1975},
doi = {10.1007/bf01437217},
}
@article{BK:77,
author = {B. K{\aa}gstr{\"o}m},
journal = {BIT Numerical Mathematics},
pages = {39-57},
title = {Bounds and perturbation bounds for the matrix
exponential},
volume = {17},
year = {1977},
abstract = {Some new types of bounds and perturbation bounds,
based on the Jordan normal form, for the matrix
exponential are derived. These bounds are compared to
known bounds, both theoretically and by numerical
examples. Some recent results on the matrix
exponential and the logarithmic norm are also
included.},
doi = {10.1007/BF01932398},
}
@article{MYL-LW:98,
author = {M. Y. Li and L. Wang},
journal = {Journal of Mathematical Analysis and Applications},
number = {1},
pages = {249-264},
title = {A Criterion for Stability of Matrices},
volume = {225},
year = {1998},
doi = {10.1006/jmaa.1998.6020},
}
@article{ZZ:03,
author = {Z. Zahreddine},
journal = {International Journal of Mathematics and Mathematical
Sciences},
note = {Article ID: 937084},
title = {Matrix measure and application to stability of
matrices and interval dynamical systems},
year = {2003},
doi = {10.1155/S0161171203202295},
}
@article{GH-ML:08,
author = {G. Hu and M. Liu},
journal = {IMA Journal of Mathematical Control and Information},
number = {1},
pages = {75-84},
title = {Properties of the weighted logarithmic matrix norms},
volume = {25},
year = {2008},
abstract = {In this paper, we are concerned with the properties
of the weighted logarithmic matrix norms. A relation
between the elliptic logarithmic matrix norm and the
weighted logarithmic matrix norm is given. Based on
Lyapunov equations, two weighted logarithmic matrix
norms are constructed which are less than
1-logarithmic matrix norm and ∞-logarithmic matrix
norm, respectively. Then, an iterative scheme is
presented to obtain the logarithmically ϵ-efficient
matrix norm. Numerical examples are given to
illustrate the results.},
doi = {10.1093/imamci/dnm006},
}
@book{AB-RJP:94,
author = {A. Berman and R. J. Plemmons},
publisher = {SIAM},
title = {Nonnegative Matrices in the Mathematical Sciences},
year = {1994},
isbn = {978-0898713213},
}
@article{XD-SJ-FB:19f,
author = {X. Duan and S. Jafarpour and F. Bullo},
journal = {SIAM Journal on Control and Optimization},
number = {5},
pages = {3447-3471},
title = {Graph-Theoretic Stability Conditions for {Metzler}
Matrices and Monotone Systems},
volume = {59},
year = {2021},
abstract = {This paper studies the graph-theoretic conditions for
stability of positive monotone systems. Using
concepts from the input-to-state stability and
network small-gain theory, we first establish
necessary and sufficient conditions for the stability
of linear positive systems described by Metzler
matrices. Specifically, we define and compute two
forms of input-to-state stability gains for Metzler
systems, namely max-interconnection gains and
sum-interconnection gains. Then, based on the
max-interconnection gains, we show that the cyclic
small-gain theorem becomes necessary and sufficient
for the stability of Metzler systems; based on the
sum-interconnection gains, we obtain novel
graph-theoretic conditions for the stability of
Metzler systems. All these conditions highlight the
role of cycles in the interconnection graph and
unveil how the structural properties of the graph
affect stability. Finally, we extend our results to
the nonlinear monotone system and obtain similar
sufficient conditions for global asymptotic
stability.},
doi = {10.1137/20M131802X},
}
@article{GGD:83,
author = {Germund Dahlquist},
journal = {Linear Algebra and its Applications},
pages = {199-216},
title = {On matrix majorants and minorants, with applications
to differential equations},
volume = {52-53},
year = {1983},
abstract = {Some tools of linear algebra are collected and
developed for potential use in the analysis of stiff
differential equations. Bounds for the triangular
factors of a large matrix are given in terms of the
triangular factors of an associated “minorant”
matrix of lower order. Minorants are also used to
produce estimates of solutions of systems of ordinary
differential equations, which may be sharper than
those obtained by the use of logarithmic norms.},
doi = {10.1016/0024-3795(83)80014-7},
}
@article{GGD:85,
author = {G. Dahlquist},
journal = {BIT Numerical Mathematics},
number = {1},
pages = {188--204},
title = {33 years of numerical instability, {Part I}},
volume = {25},
year = {1985},
doi = {10.1007/bf01934997},
}
@article{GG-SK:92,
author = {G. Giorgi and S. Koml{\'o}si},
journal = {Rivista di Matematica Per Le Scienze Economiche e
Sociali},
number = {1},
pages = {3--30},
title = {Dini derivatives in optimization --- {Part I}},
volume = {15},
year = {1992},
doi = {10.1007/BF02086523},
}
@article{FLB-JS-CW:61,
author = {F. L. Bauer and J. Stoer and C. Witzgall},
journal = {Numerische Mathematik},
pages = {257-264},
title = {Absolute and monotonic norms},
volume = {3},
year = {1961},
doi = {10.1007/BF01386026},
}
@book{DSB:09,
author = {D. S. Bernstein},
edition = {2},
publisher = {Princeton University Press},
title = {Matrix Mathematics},
year = {2009},
isbn = {0691140391},
}
@article{APM-ESP:86,
author = {A. P Molchanov and E. S. Pyatnitsky},
journal = {Automation and Remote Control},
note = {(In Russian)},
pages = {38-49},
title = {Lyapunov functions defining necessary and sufficient
conditions for the absolute stability of nonlinear
nonstationary control systems},
volume = {5},
year = {1986},
url = {http://mi.mathnet.ru/eng/at6230},
}
@article{HK-JA-PS:92,
author = {H. {Kiendl} and J. {Adamy} and P. {Stelzner}},
journal = {IEEE Transactions on Automatic Control},
number = {6},
pages = {839-842},
title = {Vector norms as {Lyapunov} functions for linear
systems},
volume = {37},
year = {1992},
doi = {10.1109/9.256362},
}
@article{AP:97,
author = {A. {Polanski}},
journal = {IEEE Transactions on Automatic Control},
number = {7},
pages = {1013-1016},
title = {Lyapunov function construction by linear programming},
volume = {42},
year = {1997},
doi = {10.1109/9.599986},
}
@article{KL-AP-RR:98,
author = {K. Loskot and A. Polanski and R. Rudnicki},
journal = {IEEE Transactions on Automatic Control},
number = {2},
pages = {289-291},
title = {{Further comments on "Vector norms as {Lyapunov}
functions for linear systems"}},
volume = {43},
year = {1998},
doi = {10.1109/9.661083},
}
@article{CM-CVL:03,
author = {C. Moler and C. V. Loan},
journal = {SIAM Review},
number = {1},
pages = {3-49},
title = {Nineteen Dubious Ways to Compute the Exponential of a
Matrix, Twenty-Five Years Later},
volume = {45},
year = {2003},
doi = {10.1137/S00361445024180},
}
@article{CRJ:75,
author = {C. R. Johnson},
journal = {Journal of Research of the National Bureau of
Standards - B. Mathematical Sciences},
pages = {97--102},
title = {Two submatrix properties of certain induced norms},
volume = {79},
year = {1975},
url = {https://nvlpubs.nist.gov/nistpubs/jres/79B/jresv79Bn3-
4p97_A1b.pdf},
}
@article{MF-VP:62,
author = {M. Fiedler and V. Ptak},
journal = {Czechoslovak Mathematical Journal},
number = {3},
pages = {382--400},
title = {On matrices with non-positive off-diagonal elements
and positive principal minors},
volume = {12},
year = {1962},
doi = {10.21136/CMJ.1962.100526},
}
@inproceedings{MJT:04,
author = {M. J. Tsatsomeros},
booktitle = {Focus on Computational Neurobiology},
editor = {Lei Li},
pages = {115-132},
publisher = {Nova Science Publishers},
title = {Generating and Detecting Matrices with Positive
Principal Minors},
year = {2004},
abstract = {A brief but concise review of methods to generate
P-matrices (i.e., matrices having positive principal
minors) is provided and motivated by open problems on
P-matrices and the desire to develop and test
efficient methods for the detection of P-matrices.
Also discussed are operations that leave the class of
P-matrices invariant. Some new results and extensions
of results regarding P-matrices are included.},
isbn = {1590339150},
}
@unpublished{SJ:19-personal,
author = {S. Jafarpour},
note = {Personal communication},
title = {A sign contractivity property of {Metzler} matrices},
year = {2019},
}
@article{HM-SK-YO:78,
author = {H. Maeda and S. Kodama and Y. Ohta},
journal = {IEEE Transactions on Circuits and Systems},
number = {6},
pages = {372-378},
title = {Asymptotic behavior of nonlinear compartmental
systems: {N}onoscillation and stability},
volume = {25},
year = {1978},
doi = {10.1109/TCS.1978.1084490},
}
@article{JAJ-CPS:93,
author = {J. A. Jacquez and C. P. Simon},
journal = {SIAM Review},
number = {1},
pages = {43--79},
title = {Qualitative theory of compartmental systems},
volume = {35},
year = {1993},
doi = {10.1137/1035003},
}
@article{HHR:63,
author = {H. H. Rosenbrock},
journal = {Automatica},
number = {1},
pages = {31--53},
title = {A {Lyapunov} function with applications to some
nonlinear physical systems},
volume = {1},
year = {1963},
doi = {10.1016/0005-1098(63)90005-0},
}
@inproceedings{SPB-DSB:99,
author = {S. P. Bhat and D. S. Bernstein},
booktitle = {{A}merican {C}ontrol {C}onference},
pages = {1608--1612},
title = {Lyapunov analysis of semistability},
volume = {3},
year = {1999},
doi = {10.1109/ACC.1999.786101},
}
@article{LM-HY:1960,
author = {L. Markus and H. Yamabe},
journal = {Osaka Mathematical Journal},
number = {2},
pages = {305--317},
publisher = {Osaka University and Osaka City University,
Departments of Mathematics},
title = {Global stability criteria for differential systems},
volume = {12},
year = {1960},
doi = {10.18910/9397},
}
@book{JCD-BAF-ART:90,
author = {J. C. Doyle and B. A. Francis and A. R. Tannenbaum},
publisher = {MacMillan Publishing Co},
title = {Feedback Control Theory},
year = {1990},
isbn = {0486469336},
}
@book{AFF:88,
author = {A. F. Filippov},
publisher = {Kluwer},
title = {Differential Equations with Discontinuous Righthand
Sides},
year = {1988},
isbn = {902772699X},
}
@article{RAS:86,
author = {Smith, R. A.},
journal = {Proceedings of the Royal Society of Edinburgh Section
A: Mathematics},
number = {3-4},
pages = {235--259},
title = {Some applications of {Hausdorff} dimension
inequalities for ordinary differential equations},
volume = {104},
year = {1986},
doi = {10.1017/S030821050001920X},
}
@article{WL-TC:06,
author = {W. Lu and T. Chen},
journal = {Physica D: Nonlinear Phenomena},
number = {2},
pages = {214-230},
title = {New approach to synchronization analysis of linearly
coupled ordinary differential systems},
volume = {213},
year = {2006},
abstract = {In this paper, a general framework is presented for
analyzing the synchronization stability of Linearly
Coupled Ordinary Differential Equations (LCODEs). The
uncoupled dynamical behavior at each node is general,
and can be chaotic or otherwise; the coupling
configuration is also general, with the coupling
matrix not assumed to be symmetric or irreducible. On
the basis of geometrical analysis of the
synchronization manifold, a new approach is proposed
for investigating the stability of the
synchronization manifold of coupled oscillators. In
this way, criteria are obtained for both local and
global synchronization. These criteria indicate that
the left and right eigenvectors corresponding to
eigenvalue zero of the coupling matrix play key roles
in the stability analysis of the synchronization
manifold. Furthermore, the roles of the uncoupled
dynamical behavior on each node and the coupling
configuration in the synchronization process are also
studied.},
doi = {10.1016/j.physd.2005.11.009},
}
@article{TC-PEK:05,
author = {T. Caraballo and P. E. Kloeden},
journal = {Proceedings of the Royal Society A: Mathematical,
Physical and Engineering Sciences},
number = {2059},
pages = {2257-2267},
title = {The persistence of synchronization under
environmental noise},
volume = {461},
year = {2005},
abstract = {It is shown that the synchronization of dissipative
systems persists when they are disturbed by additive
noise, no matter how large the intensity of the
noise, provided asymptotically stable
stationary-stochastic solutions are used instead of
asymptotically stable equilibria.},
doi = {10.1098/rspa.2005.1484},
}
@article{LDA-MC:13,
author = {L. D'Alto and M. Corless},
journal = {Numerical Algebra, Control and Optimization},
pages = {175-201},
title = {Incremental quadratic stability},
volume = {3},
year = {2013},
abstract = {The concept of incremental quadratic stability
($\delta$QS) is very useful in treating systems with
persistently acting inputs. To illustrate, if a
time-invariant $\delta$QS system is subject to a
constant input or $T$-periodic input then, all its
trajectories exponentially converge to a unique
constant or $T$-periodic trajectory, respectively. By
considering the relationship of $\delta$QS to the
usual concept of quadratic stability, we obtain a
useful necessary and sufficient condition for
$\delta$QS. A main contribution of the paper is to
consider nonlinear/uncertain systems whose state
dependent nonlinear/uncertain terms satisfy an
incremental quadratic constraint which is
characterized by a bunch of symmetric matrices we
call incremental multiplier matrices. We obtain
linear matrix inequalities whose feasibility
guarantee $\delta$QS of these systems. Frequency
domain characterizations of $\delta$QS are then
obtained from these conditions. By characterizing
incremental multiplier matrices for many common
classes of nonlinearities, we demonstrate the
usefulness of our results.},
doi = {10.3934/naco.2013.3.175},
}
@article{DCL:51,
author = {Lewis, D. C.},
journal = {American Journal of Mathematics},
number = {1},
pages = {48-58},
title = {Differential equations referred to a variable metric},
volume = {73},
year = {1951},
doi = {10.2307/2372159},
}
@book{TY:1966,
author = {T. Yoshizawa},
publisher = {Mathematical Society of Japan},
title = {Stability Theory by {Liapunov}'s Second Method},
year = {1966},
}
@article{VF-SM-DNC:96,
author = {V. Fromion and S. Monaco and D. Normand-Cyrot},
journal = {{IEEE} Transactions on Automatic Control},
number = {5},
pages = {721--723},
title = {Asymptotic properties of incrementally stable
systems},
volume = {41},
year = {1996},
doi = {10.1109/9.489210},
}
@article{DA:02,
author = {D. Angeli},
journal = {IEEE Transactions on Automatic Control},
number = {3},
pages = {410--421},
title = {A {Lyapunov} approach to incremental stability
properties},
volume = {47},
year = {2002},
abstract = {Deals with several notions of incremental stability.
In other words, the focus is on stability of
trajectories with respect to one another, rather than
with respect to some attractor. The aim is to present
a framework for understanding such questions fully
compatible with the well-known input-to-state
stability approach. Applications of the newly
introduced stability notions are also discussed.},
doi = {10.1109/9.989067},
}
@article{GJM:62,
author = {G. J. Minty},
journal = {Duke Mathematical Journal},
number = {3},
pages = {341-346},
title = {Monotone {(nonlinear)} operators in {H}ilbert space},
volume = {29},
year = {1962},
doi = {10.1215/S0012-7094-62-02933-2},
}
@article{GJM:64,
author = {G. J. Minty},
journal = {Pacific Journal of Mathematics},
number = {1},
pages = {243-247},
title = {On the monotonicity of the gradient of a convex
function},
volume = {14},
year = {1964},
doi = {10.2140/pjm.1964.14.243},
}
@misc{SC-MA:13,
author = {S. Coogan and M. Arcak},
title = {A note on norm-based {Lyapunov} functions via
contraction analysis},
year = {2013},
url = {https://arxiv.org/pdf/1308.0586},
}
@article{RAS:86b,
author = {R. A Smith},
journal = {Journal of Mathematical Analysis and Applications},
number = {2},
pages = {679--708},
title = {Massera{\textquotesingle}s convergence theorem for
periodic nonlinear differential equations},
volume = {120},
year = {1986},
doi = {10.1016/0022-247x(86)90189-7},
}
@article{JJES-WL:01,
author = {J.-J. E. Slotine and Lohmiller, W.},
journal = {Neural Networks},
number = {2},
pages = {137--145},
title = {Modularity, evolution, and the binding problem: a
view from stability theory},
volume = {14},
year = {2001},
doi = {10.1016/S0893-6080(00)00089-7},
}
@article{MAAR-DA:16,
author = {M. A. Al-Radhawi and D. {Angeli}},
journal = {IEEE Transactions on Automatic Control},
number = {1},
pages = {76-89},
title = {New Approach to the Stability of Chemical Reaction
Networks: {Piecewise} Linear in Rates {Lyapunov}
Functions},
volume = {61},
year = {2016},
doi = {10.1109/TAC.2015.2427691},
}
@article{SC-MA:15,
author = {S. Coogan and M. Arcak},
journal = {IEEE Transactions on Automatic Control},
number = {10},
pages = {2698--2703},
title = {A compartmental model for traffic networks and its
dynamical behavior},
volume = {60},
year = {2015},
abstract = {We propose a macroscopic traffic network flow model
suitable for analysis as a dynamical system, and we
qualitatively analyze equilibrium flows as well as
convergence. Flows at a junction are determined by
downstream supply of capacity as well as upstream
demand of traffic wishing to flow through the
junction. This approach is rooted in the celebrated
Cell Transmission Model for freeway traffic flow.
Unlike related results which rely on certain system
cooperativity properties, our model generally does
not possess these properties. We show that the lack
of cooperativity is in fact a useful feature that
allows traffic control methods, such as ramp
metering, to be effective. Finally, we leverage the
results of the technical note to develop a linear
program for optimal ramp metering.},
doi = {10.1109/TAC.2015.2411916},
}
@article{GC-EL-KS:15,
author = {G. Como and E. Lovisari and K. Savla},
journal = {IEEE Transactions on Control of Network Systems},
number = {1},
pages = {57-67},
title = {Throughput Optimality and Overload Behavior of
Dynamical Flow Networks Under Monotone Distributed
Routing},
volume = {2},
year = {2015},
doi = {10.1109/TCNS.2014.2367361},
}
@article{JJH:1982,
author = {J. J. Hopfield},
journal = {Proceedings of the National Academy of Sciences},
number = {8},
pages = {2554--2558},
title = {Neural networks and physical systems with emergent
collective computational abilities},
volume = {79},
year = {1982},
abstract = {Computational properties of use of biological
organisms or to the construction of computers can
emerge as collective properties of systems having a
large number of simple equivalent components (or
neurons). The physical meaning of content-addressable
memory is described by an appropriate phase space
flow of the state of a system. A model of such a
system is given, based on aspects of neurobiology but
readily adapted to integrated circuits. The
collective properties of this model produce a
content-addressable memory which correctly yields an
entire memory from any subpart of sufficient size.
The algorithm for the time evolution of the state of
the system is based on asynchronous parallel
processing. Additional emergent collective properties
include some capacity for generalization, familiarity
recognition, categorization, error correction, and
time sequence retention. The collective properties
are only weakly sensitive to details of the modeling
or the failure of individual devices.},
doi = {10.1073/pnas.79.8.2554},
}
@article{KDM-FF:12,
author = {K. D. Miller and F. Fumarola},
journal = {Neural Computation},
number = {1},
pages = {25-31},
title = {Mathematical Equivalence of Two Common Forms of
Firing Rate Models of Neural Networks},
volume = {24},
year = {2012},
doi = {10.1162/NECO_a_00221},
}
@article{JJH:84,
author = {J. J. Hopfield},
journal = {Proceedings of the National Academy of Sciences},
number = {10},
pages = {3088-3092},
title = {Neurons with graded response have collective
computational properties like those of two-state
neurons},
volume = {81},
year = {1984},
abstract = {A model for a large network of "neurons" with a
graded response (or sigmoid input-output relation) is
studied. This deterministic system has collective
properties in very close correspondence with the
earlier stochastic model based on McCulloch - Pitts
neurons. The content- addressable memory and other
emergent collective properties of the original model
also are present in the graded response model. The
idea that such collective properties are used in
biological systems is given added credence by the
continued presence of such properties for more nearly
biological "neurons." Collective analog electrical
circuits of the kind described will certainly
function. The collective states of the two models
have a simple correspondence. The original model will
continue to be useful for simulations, because its
connection to graded response systems is established.
Equations that include the effect of action
potentials in the graded response system are also
developed.},
doi = {10.1073/pnas.81.10.3088},
}
@article{ANM-JAF-WP:89,
author = {A. N. {Michel} and J. A. {Farrell} and W. {Porod}},
journal = {IEEE Transactions on Circuits and Systems},
number = {2},
pages = {229-243},
title = {Qualitative analysis of neural networks},
volume = {36},
year = {1989},
doi = {10.1109/31.20200},
}
@article{EK-AB:94,
author = {Kaszkurewicz, E. and Bhaya, A.},
journal = {IEEE Transactions on Circuits and Systems I:
Fundamental Theory and Applications},
number = {2},
pages = {171-174},
title = {On a class of globally stable neural circuits},
volume = {41},
year = {1994},
abstract = {The authors show that diagonal stability of the
interconnection matrix leads to a simple proof of the
existence, uniqueness, and global asymptotic
stability of the equilibrium of a Hopfield-Tank
neural circuit, without making some common
restrictive assumptions used in earlier results. It
is also shown that the same condition guarantees
structural stability, which ensures the desirable
property of persistence of global asymptotic
stability under general C/sup 1/
perturbations.<>},
doi = {10.1109/81.269055},
}
@article{MF-SM-MM:94,
author = {Forti, M. and Manetti, S. and Marini, M.},
journal = {IEEE Transactions on Circuits and Systems I:
Fundamental Theory and Applications},
number = {7},
pages = {491-494},
title = {Necessary and sufficient condition for absolute
stability of neural networks},
volume = {41},
year = {1994},
abstract = {The main result in this paper is that for a neural
circuit of the Hopfield type with a symmetric
connection matrix T, the negative semidefiniteness of
T is a necessary and sufficient condition for
Absolute Stability. The most significant theoretical
implication is that the class of neural circuits with
a negative semidefinite T is the largest class of
circuits that can be employed for embedding and
solving optimization problems without the risk of
spurious responses.},
doi = {10.1109/81.298364},
}
@article{MF-AT:95,
author = {M. Forti and A. Tesi},
journal = {IEEE Transactions on Circuits and Systems I:
Fundamental Theory and Applications},
number = {7},
pages = {354-366},
title = {New conditions for global stability of neural
networks with application to linear and quadratic
programming problems},
volume = {42},
year = {1995},
abstract = {In this paper, we present new conditions ensuring
existence, uniqueness, and Global Asymptotic
Stability (GAS) of the equilibrium point for a large
class of neural networks. The results are applicable
to both symmetric and nonsymmetric interconnection
matrices and allow for the consideration of all
continuous nondecreasing neuron activation functions.
Such functions may be unbounded (but not necessarily
surjective), may have infinite intervals with zero
slope as in a piece-wise-linear model, or both. The
conditions on GAS rely on the concept of Lyapunov
Diagonally Stable (or Lyapunov Diagonally
Semi-Stable) matrices and are proved by employing a
class of Lyapunov functions of the generalized
Lur'e-Postnikov type. Several classes of
interconnection matrices of applicative interest are
shown to satisfy our conditions for GAS. In
particular, the results are applied to analyze GAS
for the class of neural circuits introduced for
solving linear and quadratic programming problems. In
this application, the principal result here obtained
is that these networks are GAS also when the
constraint amplifiers are dynamical, as it happens in
any practical implementation.<>},
doi = {10.1109/81.401145},
}
@article{SA:02,
author = {Arik, S.},
journal = {IEEE Transactions on Circuits and Systems I:
Fundamental Theory and Applications},
number = {4},
pages = {502-504},
title = {A note on the global stability of dynamical neural
networks},
volume = {49},
year = {2002},
abstract = {It is shown that the additive diagonal stability
condition on the interconnection matrix of a neural
network, together with the assumption that the
activation functions are nondecreasing, guarantees
the uniqueness of the equilibrium point. This
condition, under the same assumption on the
activation functions, is also shown to imply the
local attractivity and local asymptotic stability of
the equilibrium point, thus ensuring the global
asymptotic stability (GAS) of the equilibrium point.
The result obtained generalizes the previous results
derived in the literature.},
doi = {10.1109/81.995665},
}
@article{HZ-ZW-DL:14,
author = {H. {Zhang} and Z. {Wang} and D. {Liu}},
journal = {IEEE Transactions on Neural Networks and Learning
Systems},
number = {7},
pages = {1229-1262},
title = {A Comprehensive Review of Stability Analysis of
Continuous-Time Recurrent Neural Networks},
volume = {25},
year = {2014},
doi = {10.1109/TNNLS.2014.2317880},
}
@article{MR-RW-IRM:20,
author = {M. Revay and R. Wang and I. R. Manchester},
title = {Lipschitz Bounded Equilibrium Networks},
year = {2020},
url = {https://arxiv.org/abs/2010.01732},
}
@misc{LK-ME-JJES:21,
author = {L. Kozachkov and M. Ennis and J.-J. E. Slotine},
title = {{RNNs} of {RNNs}: {Recursive} Construction of Stable
Assemblies of Recurrent Neural Networks},
year = {2021},
url = {https://arxiv.org/abs/2106.08928},
}
@article{MAK-SGK:1955,
author = {M. A. Krasnoselski{\"\i} and S. G. Krein},
booktitle = {Dokl. Akad. Nauk SSSR},
number = {1},
pages = {13--16},
title = {Nonlocal existence theorems and uniqueness theorems
for systems of ordinary differential equations},
volume = {102},
year = {1955},
}
@book{RPA-VL:93,
author = {R. P. Agarwal and V. Lakshmikantham},
publisher = {World Scientific},
title = {Uniqueness and nonuniqueness criteria for ordinary
differential equations},
year = {1993},
}
@book{KD:85,
author = {K. Deimling},
publisher = {Springer},
title = {Nonlinear Functional Analysis},
year = {1985},
isbn = {3-540-13928-1},
}
@article{JL-XL-WCX-HZ:11,
author = {J. Liu and X. Liu and W.-C. Xie and H. Zhang},
journal = {Automatica},
number = {12},
pages = {2689-2696},
title = {Stochastic consensus seeking with communication
delays},
volume = {47},
year = {2011},
abstract = {This paper investigates the consensus problem of
dynamical networks of multi-agents where each agent
can only obtain noisy and delayed measurements of the
states of its neighbors due to environmental
uncertainties and communication delays. We consider
general networks with fixed topology and with
switching (dynamically changing) topology, propose
consensus protocols that take into account both the
noisy measurements and the communication time-delays,
and study mean square average-consensus for
multi-agent systems networked in an uncertain
environment and with uniform communication
time-varying delays. Using tools from differential
equations and stochastic calculus, together with
results from matrix theory and algebraic graph
theory, we establish sufficient conditions under
which the proposed consensus protocols lead to mean
square average-consensus. Simulations are also
provided to demonstrate the theoretical results.},
doi = {10.1016/j.automatica.2011.09.005},
}
@book{AH:1966,
author = {A. Halanay},
publisher = {Academic Press},
title = {Differential Equations: Stability, Oscillations, Time
Lags},
year = {1966},
isbn = {978-0-12-317950-0},
}
@book{RA-JEM-TSR:88,
author = {R. Abraham and J. E. Marsden and T. S. Ratiu},
edition = {2},
publisher = {Springer},
series = {Applied Mathematical Sciences},
title = {Manifolds, Tensor Analysis, and Applications},
volume = {75},
year = {1988},
isbn = {0387967907},
}
@article{HC-XL-WZ-YC:16,
author = {H. Chu and X. Liu and W. Zhang and Y. Cai},
journal = {Journal of the Franklin Institute},
number = {7},
pages = {1594-1614},
title = {Observer-based consensus tracking of multi-agent
systems with one-sided {Lipschitz} nonlinearity},
volume = {353},
year = {2016},
abstract = {This paper investigates the observer-based consensus
tracking problem of multi-agent systems with
one-sided Lipschitz nonlinearity. The agent dynamics
considered here covers a broad family of nonlinear
systems, and includes the well-known Lipschitz system
as a special case. To achieve consensus tracking for
such multi-agent systems, two types of observer-based
protocols named the continuous protocol and the
intermittent protocol are proposed. Furthermore,
several multi-step design algorithms are presented to
select the observer gains and the controller
parameters of the proposed protocols. It is shown
that the established sufficient criteria can not only
ensure the observer error to approach to zero, but
also realize the consensus tracking of multi-agent
systems. The obtained results are illustrated by two
simulation examples.},
doi = {10.1016/j.jfranklin.2015.10.011},
}
@article{MN:1942,
author = {M. Nagumo},
journal = {Proceedings of the Physico-Mathematical Society of
Japan. 3rd Series},
pages = {551-559},
title = {{\"Uber die Lage der Integralkurven gew\"ohnlicher
Differentialgleichungen}},
volume = {24},
year = {1942},
doi = {10.11429/ppmsj1919.24.0_551},
}
@article{FB:99,
author = {F. Blanchini},
journal = {Automatica},
number = {11},
pages = {1747-1767},
title = {Set invariance in control},
volume = {35},
year = {1999},
doi = {10.1016/S0005-1098(99)00113-2},
}
@book{FB-SM:15,
author = {F. Blanchini and S. Miani},
publisher = {Springer},
title = {Set-Theoretic Methods in Control},
year = {2015},
isbn = {9783319179322},
}
@article{JAY:69,
author = {J. A. Yorke},
journal = {Proceedings of the American Mathematical Society},
number = {2},
pages = {509--512},
title = {Periods of periodic solutions and the {Lipschitz}
constant},
volume = {22},
year = {1969},
doi = {10.2307/2037090},
}
@article{AL-JAY:76,
author = {A. Lajmanovich and J. A. Yorke},
journal = {Mathematical Biosciences},
number = {3},
pages = {221--236},
title = {A deterministic model for gonorrhea in a
nonhomogeneous population},
volume = {28},
year = {1976},
doi = {10.1016/0025-5564(76)90125-5},
}
@article{IWS:78,
author = {I. W. Sandberg},
journal = {IEEE Transactions on Circuits and Systems},
number = {5},
pages = {273-279},
title = {On the mathematical foundations of compartmental
analysis in biology, medicine, and ecology},
volume = {25},
year = {1978},
doi = {10.1109/TCS.1978.1084473},
}
@article{RIK:60,
author = {R. I. Kachurovskii},
journal = {Uspekhi Matematicheskikh Nauk},
number = {4},
pages = {213--215},
publisher = {Russian Academy of Sciences, Steklov Mathematical
Institute of Russian~…},
title = {Monotone operators and convex functionals},
volume = {15},
year = {1960},
}
@book{DB-AN-AO:03,
author = {D. Bertsekas and A. Nedi{\'c} and A. Ozdaglar},
publisher = {Athena Scientific},
title = {Convex Analysis and Optimization},
year = {2003},
isbn = {1-886529-45-0},
}
@article{EKR-SB:16,
author = {E. K. Ryu and S. Boyd},
journal = {Applied Computational Mathematics},
number = {1},
pages = {3--43},
title = {Primer on monotone operator methods},
volume = {15},
year = {2016},
}
@book{EPO:59,
author = {E. P. Odum},
publisher = {Saunders Company},
title = {Fundamentals of Ecology},
year = {1959},
}
@article{WM-SM-SZ-FB:16f,
author = {W. Mei and S. Mohagheghi and S. Zampieri and
F. Bullo},
journal = {Annual Reviews in Control},
pages = {116-128},
title = {On the Dynamics of Deterministic Epidemic Propagation
over Networks},
volume = {44},
year = {2017},
abstract = {In this work we review a class of deterministic
nonlinear models for the propagation of infectious
diseases over contact networks with
strongly-connected topologies. We consider network
models for susceptible-infected (SI),
susceptible-infected-susceptible (SIS), and
susceptible-infected-recovered (SIR) settings. In
each setting, we provide a comprehensive nonlinear
analysis of equilibria, stability properties,
convergence, monotonicity, positivity, and threshold
conditions. For the network SI setting, specific
contributions include establishing its equilibria,
stability, and positivity properties. For the network
SIS setting, we review a well- known deterministic
model, provide novel results on the computation and
characterization of the endemic state (when the
system is above the epidemic threshold), and present
alternative proofs for some of its properties.
Finally, for the network SIR setting, we propose
novel results for transient behavior, threshold
conditions, stability properties, and asymptotic
convergence. These results are analogous to those
well-known for the scalar case. In addition, we
provide a novel iterative algorithm to compute the
asymptotic state of the network SIR system.},
doi = {10.1016/j.arcontrol.2017.09.002},
}
@article{CFD:94,
author = {C. F. Daganzo},
journal = {Transportation Research Part B: Methodological},
number = {4},
pages = {269-287},
title = {The cell transmission model: A dynamic representation
of highway traffic consistent with the hydrodynamic
theory},
volume = {28},
year = {1994},
abstract = {This paper presents a simple representation of
traffic on a highway with a single entrance and exit.
The representation can be used to predict traffic's
evolution over time and space, including transient
phenomena such as the building, propagation, and
dissipation of queues. The easy-to-solve difference
equations used to predict traffic's evolution are
shown to be the discrete analog of the differential
equations arising from a special case of the
hydrodynamic model of traffic flow. The proposed
method automatically generates appropriate changes in
density at locations where the hydrodynamic theory
would call for a shockwave; i.e., a jump in density
such as those typically seen at the end of every
queue. The complex side calculations required by
classical methods to keep track of shockwaves are
thus eliminated. The paper also shows how the
equations can mimic the real-life development of
stop-and-go traffic within moving queues.},
doi = {10.1016/0191-2615(94)90002-7},
}
@inproceedings{EL-GC-KS:14,
address = {Los Angeles, USA},
author = {E. {Lovisari} and G. {Como} and K. {Savla}},
booktitle = {{IEEE} Conf.\ on Decision and Control},
month = dec,
pages = {2384-2389},
title = {Stability of monotone dynamical flow networks},
year = {2014},
doi = {10.1109/CDC.2014.7039752},
}
@article{SC-MA:16,
author = {S. Coogan and M. Arcak},
journal = {Automatica},
pages = {246-253},
title = {Stability of traffic flow networks with a polytree
topology},
volume = {66},
year = {2016},
doi = {10.1016/j.automatica.2015.12.015},
}
@inproceedings{SC-MA:15b,
author = {S. Coogan and M. Arcak},
booktitle = {Hybrid Systems: Computation and Control},
month = apr,
pages = {58--67},
title = {Efficient finite abstraction of mixed monotone
systems},
year = {2015},
abstract = {We present an efficient computational procedure for
finite abstraction of discrete-time mixed monotone
systems by considering a rectangular partition of the
state space. Mixed monotone systems are decomposable
into increasing and decreasing components, and
significantly generalize the well known class of
monotone systems. We tightly overapproximate the
one-step reachable set from a box of initial
conditions by computing a decomposition function at
only two points, regardless of the dimension of the
state space. We apply our results to verify the
dynamical behavior of a model for insect population
dynamics and to synthesize a signaling strategy for a
traffic network.},
doi = {10.1145/2728606.2728607},
}
@article{JM:87,
author = {J. Mierczyński},
journal = {SIAM Journal on Mathematical Analysis},
number = {3},
pages = {642-646},
title = {Strictly Cooperative Systems with a First Integral},
volume = {18},
year = {1987},
doi = {10.1137/0518049},
}
@article{DA-EDS:08,
author = {D. Angeli and E. D. Sontag},
journal = {Nonlinear Analysis: Real World Applications},
number = {1},
pages = {128-140},
title = {Translation-invariant monotone systems, and a global
convergence result for enzymatic futile cycles},
volume = {9},
year = {2008},
abstract = {Strongly monotone systems of ordinary differential
equations which have a certain translation-invariance
property are shown to have the property that all
projected solutions converge to a unique equilibrium.
This result may be seen as a dual of a well-known
theorem of Mierczyński for systems that satisfy a
conservation law. As an application, it is shown that
enzymatic futile cycles have a global convergence
property.},
doi = {10.1016/j.nonrwa.2006.09.006},
}
@article{VSB-KS:97,
author = {Borkar, V. S. and Soumyanatha, K.},
journal = {IEEE Transactions on Circuits and Systems I:
Fundamental Theory and Applications},
number = {4},
pages = {351-355},
title = {An analog scheme for fixed point computation. {I.}
{Theory}},
volume = {44},
year = {1997},
abstract = {An analog system for fixed point computation is
described. The system is derived from a continuous
time analog of the classical over-relaxed fixed point
iteration. The dynamical system is proved to converge
for nonexpansive mappings under all p norms, p/spl
isin/(1,/spl infin/). This extends previously
established results to not necessarily differentiable
maps which are nonexpansive under the /spl
infin/-norm. The system will always converge to a
single fixed point in a connected set of fixed
points. This allows the system to function as a
complementary paradigm to energy minimization
techniques for optimization in the analog domain. It
is shown that the proposed technique is applicable to
a large class of dynamic programming computations.},
doi = {10.1109/81.563625},
}
@article{TY-XY-JW-DW:19,
author = {T. Yang and X. Yi and J. Wu and Y. Yuan and D. Wu and
Z. Meng and Y. Hong and H. Wang and Z. Lin and
K. H. Johansson},
journal = {Annual Reviews in Control},
pages = {278-305},
title = {A survey of distributed optimization},
volume = {47},
year = {2019},
doi = {10.1016/j.arcontrol.2019.05.006},
}
@article{DF-FP:10,
author = {D. Feijer and F. Paganini},
journal = {Automatica},
number = {12},
pages = {1974--1981},
title = {Stability of primal--dual gradient dynamics and
applications to network optimization},
volume = {46},
year = {2010},
doi = {10.1016/j.automatica.2010.08.011},
}
@article{GQ-NL:19,
author = {G. {Qu} and N. {Li}},
journal = {IEEE Control Systems Letters},
number = {1},
pages = {43-48},
title = {On the Exponential Stability of Primal-Dual Gradient
Dynamics},
volume = {3},
year = {2019},
doi = {10.1109/LCSYS.2018.2851375},
}
@inproceedings{JW-NE:11,
address = {Orlando, USA},
author = {J. Wang and N. Elia},
booktitle = {{IEEE} Conf.\ on Decision and Control and European
Control Conference},
pages = {3800-3805},
title = {A control perspective for centralized and distributed
convex optimization},
year = {2011},
doi = {10.1109/CDC.2011.6161503},
}
@article{AJL:1920,
author = {A. J. Lotka},
journal = {Proceedings of the National Academy of Sciences},
number = {7},
pages = {410-415},
title = {Analytical note on certain rhythmic relations in
organic systems},
volume = {6},
year = {1920},
doi = {10.1073/pnas.6.7.410},
}
@article{VV:1928,
author = {V. Volterra},
journal = {ICES Journal of Marine Science},
number = {1},
pages = {3--51},
publisher = {Oxford University Press},
title = {Variations and fluctuations of the number of
individuals in animal species living together},
volume = {3},
year = {1928},
doi = {10.1093/icesjms/3.1.3},
}
@article{SJ-PCV-FB:19q+arxiv,
author = {S. Jafarpour and P. Cisneros-Velarde and F. Bullo},
note = {Extended report with proofs.},
title = {Weak and Semi-Contraction for Network Systems and
Diffusively-Coupled Oscillators},
year = {2021},
url = {https://arxiv.org/abs/2005.09774},
}
@article{BSG:76,
author = {B. S. Goh},
journal = {Journal of Mathematical Biology},
number = {3-4},
pages = {313--318},
title = {Global stability in two species interactions},
volume = {3},
year = {1976},
doi = {10.1007/BF00275063},
}
@article{YT-NA-HT:78,
author = {Y. Takeuchi and N. Adachi and H. Tokumaru},
journal = {Journal of Mathematical Analysis and Applications},
number = {3},
pages = {453--473},
title = {The stability of generalized {Volterra} equations},
volume = {62},
year = {1978},
doi = {10.1016/0022-247X(78)90139-7},
}
@article{BSG:79,
author = {B. S. Goh},
journal = {American Naturalist},
pages = {261--275},
title = {Stability in models of mutualism},
year = {1979},
doi = {10.1086/283384},
}
@book{BSG:80,
author = {B.-S. Goh},
publisher = {Elsevier},
title = {Management and Analysis of Biological Populations},
year = {1980},
isbn = {978-0-444-41793-0},
}
@book{YT:96,
author = {Y. Takeuchi},
publisher = {World Scientific Publishing},
title = {Global Dynamical Properties of {Lotka-Volterra}
Systems},
year = {1996},
isbn = {9810224710},
}
@unpublished{SB:10,
author = {S. Baigent},
month = mar,
note = {Unpublished Lecture Notes, University of College,
London},
title = {{Lotka-Volterra Dynamics \textemdash{} An
Introduction}},
year = {2010},
annote = {Downloaded on 12/23/2016},
url = {http://www.ltcc.ac.uk/media/london-taught-course-centre/
documents/Bio-Mathematics-(APPLIED).pdf},
}
@book{JH-KS:98,
author = {J. Hofbauer and K. Sigmund},
publisher = {Cambridge University Press},
title = {Evolutionary Games and Population Dynamics},
year = {1998},
isbn = {052162570X},
}
@book{WHS:10,
author = {W. H. Sandholm},
publisher = {MIT Press},
title = {Population Games and Evolutionary Dynamics},
year = {2010},
isbn = {0262195879},
}
@article{MM:1927,
author = {M. M{\"u}ller},
journal = {Mathematische Zeitschrift},
number = {1},
pages = {619--645},
title = {{{\"U}ber das Fundamentaltheorem in der Theorie der
gew{\"o}hnlichen Differentialgleichungen}},
volume = {26},
year = {1927},
doi = {10.1007/BF01475477},
}
@article{EK:1932,
author = {E. Kamke},
journal = {Acta Mathematica},
pages = {57-85},
title = {{Zur Theorie der Systeme gewöhnlicher
Differentialgleichungen. II.}},
volume = {58},
year = {1932},
doi = {10.1007/BF02547774},
}
@article{MWH:82,
author = {M. W. Hirsch},
journal = {{SIAM} Journal on Mathematical Analysis},
number = {2},
pages = {167--179},
title = {Systems of Differential Equations Which Are
Competitive or Cooperative: {I}. {Limit} Sets},
volume = {13},
year = {1982},
doi = {10.1137/0513013},
}
@article{MWH:85,
author = {M. W. Hirsch},
journal = {SIAM Journal on Mathematical Analysis},
number = {3},
pages = {423--439},
title = {Systems of differential equations that are
competitive or cooperative {II}: {C}onvergence almost
everywhere},
volume = {16},
year = {1985},
doi = {10.1137/0516030},
}
@article{MWS:88,
author = {M. W. Hirsch},
journal = {Nonlinearity},
number = {1},
pages = {51--71},
title = {Systems of differential equations which are
competitive or cooperative: {III}. {Competing}
species},
volume = {1},
year = {1988},
doi = {10.1088/0951-7715/1/1/003},
}
@article{HLS:88,
author = {H. L. Smith},
journal = {SIAM Review},
number = {1},
pages = {87--113},
title = {Systems of ordinary differential equations which
generate an order preserving flow. {A} survey of
results},
volume = {30},
year = {1988},
doi = {10.1137/1030003},
}
@book{HLS:95,
author = {H. L. Smith},
publisher = {American Mathematical Society},
title = {Monotone Dynamical Systems: An Introduction to the
Theory of Competitive and Cooperative Systems},
year = {1995},
isbn = {082180393X},
}
@inproceedings{SC:16,
address = {Las Vegas, USA},
author = {S. Coogan},
booktitle = {{IEEE} Conf.\ on Decision and Control},
month = dec,
pages = {2184-2189},
title = {Separability of {Lyapunov} functions for contractive
monotone systems},
year = {2016},
doi = {10.1109/CDC.2016.7798587},
}
@article{GG-RH-AAK-PV-JK:08,
author = {G. Gomes and R. Horowitz and A. A. Kurzhanskiy and
P. Varaiya and J. Kwon},
journal = {Transportation Research Part C: Emerging
Technologies},
number = {4},
pages = {485--513},
title = {Behavior of the cell transmission model and
effectiveness of ramp metering},
volume = {16},
year = {2008},
doi = {10.1016/j.trc.2007.10.005},
}
@article{RG-AF-VS-NEL:18,
author = {R. Gray and A. Franci and V. Srivastava and
N. E. Leonard},
journal = {IEEE Transactions on Control of Network Systems},
number = {2},
pages = {793--806},
title = {Multiagent Decision-Making Dynamics Inspired by
Honeybees},
volume = {5},
year = {2018},
doi = {10.1109/tcns.2018.2796301},
}
@article{MB-GHG-JL:05,
author = {M. Benzi and G. H. Golub and J. Liesen},
journal = {Acta Numerica},
pages = {1--137},
title = {Numerical solution of saddle point problems},
volume = {14},
year = {2005},
doi = {10.1017/S0962492904000212},
}
@article{AC-BG-JC:17,
author = {A. Cherukuri and B. Gharesifard and J. Cortes},
journal = {SIAM Journal on Control and Optimization},
number = {1},
pages = {486-511},
title = {Saddle-point dynamics: {Conditions} for asymptotic
stability of saddle points},
volume = {55},
year = {2017},
abstract = {This paper considers continuously differentiable
functions of two vector variables that have (possibly
a continuum of) min-max saddle points. We study the
asymptotic convergence properties of the associated
saddle-point dynamics (gradient descent in the first
variable and gradient ascent in the second one). We
identify a suite of complementary conditions under
which the set of saddle points is asymptotically
stable under the saddle-point dynamics. Our first set
of results is based on the convexity-concavity of the
function defining the saddle-point dynamics to
establish the convergence guarantees. For functions
that do not enjoy this feature, our second set of
results relies on properties of the linearization of
the dynamics, the function along the proximal normals
to the saddle set, and the linearity of the function
in one variable. We also provide global versions of
the asymptotic convergence results. Various examples
illustrate our discussion.},
doi = {10.1137/15M1026924},
}
@article{FD-JWSP-FB:17k,
author = {F. D{\"o}rfler and J. W. Simpson-Porco and F. Bullo},
journal = {Proceedings of the IEEE},
number = {5},
pages = {977-1005},
title = {Electrical Networks and Algebraic Graph Theory:
{M}odels, Properties, and Applications},
volume = {106},
year = {2018},
abstract = {Algebraic graph theory is a cornerstone in the study
of electrical networks ranging from miniature
integrated circuits to continental-scale power
systems. Conversely, many fundamental results of
algebraic graph theory were laid out by early
electrical circuit analysts. In this paper we survey
some fundamental and historic as well as recent
results on how algebraic graph theory informs
electrical network analysis, dynamics, and design. In
particular, we review the algebraic and spectral
properties of graph adjacency, Laplacian, incidence,
and resistance matrices and how they relate to the
analysis, network-reduction, and dynamics of certain
classes of electrical networks. We study these
relations for models of increasing complexity ranging
from static resistive DC circuits, over dynamic RLC
circuits, to nonlinear AC power flow. We conclude
this paper by presenting a set of fundamental open
questions at the intersection of algebraic graph
theory and electrical networks.},
doi = {10.1109/JPROC.2018.2821924},
}
@inproceedings{ES:06-Markov,
author = {E. Seneta},
booktitle = {Markov Anniversary Meeting},
editor = {A. N. Langville and W. J. Stewart},
pages = {1--20},
publisher = {C \& M Online Media},
title = {Markov and the creation of {Markov} chains},
year = {2006},
isbn = {1-932482-34-2},
url = {https://www.csc2.ncsu.edu/conferences/nsmc},
}
@book{RAR:13,
author = {R. A. Ryan},
publisher = {Springer},
title = {Introduction to Tensor Products of Banach Spaces},
year = {2002},
isbn = {9781852334376},
}
@article{VVK:83,
author = {V. V. Kolpakov},
journal = {Journal of Soviet Mathematics},
pages = {2094–2106},
title = {Matrix seminorms and related inequalities},
volume = {23},
year = {1983},
doi = {10.1007/BF01093289},
}
@article{AAM:1906,
author = {Andrey A. Markov},
journal = {Izvestiya Fiziko-matematicheskogo obschestva pri
Kazanskom universitete},
note = {(in Russian)},
title = {Extensions of the law of large numbers to dependent
quantities},
volume = {15},
year = {1906},
}
@article{AnK:1931,
author = {A. N. Kolmogorov},
journal = {Mathematische Annalen},
pages = {415--158},
title = {{\"U}ber die analytischen {M}ethoden in der
{W}ahrscheinlichkeitsrechnung},
volume = {104},
year = {1931},
doi = {10.1007/BF01457949},
}
@article{WD:1937,
author = {W. Doeblin},
journal = {Publ. Faculty of Science University Masaryk (Brno)},
pages = {3--13},
title = {Le cas discontinu des probabilit\'es en cha{\^i}ne},
year = {1937},
issn = {0371-2125},
}
@article{RLD:1956,
author = {Dobrushin, R. L.},
journal = {Theory of Probability \& Its Applications},
number = {1},
pages = {65-80},
title = {Central Limit Theorem for Nonstationary {Markov}
Chains. {I}},
volume = {1},
year = {1956},
doi = {10.1137/1101006},
}
@book{ES:81,
author = {E. Seneta},
edition = {2},
publisher = {Springer},
title = {Non-negative Matrices and Markov Chains},
year = {1981},
isbn = {0387297650},
}
@article{RM-FJH:20,
author = {R. Marsli and F. J. Hall},
journal = {Linear and Multilinear Algebra},
number = {0},
pages = {1-21},
title = {Some properties of ergodicity coefficients with
applications in spectral graph theory},
volume = {0},
year = {2020},
abstract = {The main result is Corollary 2.9 which provides upper
bounds on, and even better, approximates the largest
non-trivial eigenvalue in absolute value of real
constant row-sum matrices by the use of vector
norm-based ergodicity coefficients {τp}. If the
constant row-sum matrix is nonsingular, then it is
also shown how its smallest non-trivial eigenvalue in
absolute value can be bounded by using {τp}. In the
last section, these two results are applied to bound
the spectral radius of the Laplacian matrix as well
as the algebraic connectivity of its associated
graph. Many other results are obtained. In
particular, Theorem 2.15 is a convergence theorem
for τp and Theorem 4.7 says that τ1 is less than
or equal to τ∞ for the Laplacian matrix of every
simple graph. An application related to the stability
of Markov chains is discussed. Other discussions,
open questions and examples are provided.},
doi = {10.1080/03081087.2020.1777251},
}
@article{JL-SM-ASM-BDOA-CY:11,
author = {J. Liu and S. Mou and A. S. Morse and
B. D. O. Anderson and C. Yu},
journal = {Proceedings of the IEEE},
number = {9},
pages = {1505-1524},
title = {Deterministic Gossiping},
volume = {99},
year = {2011},
abstract = {For the purposes of this paper, “gossiping” is a
distributed process whose purpose is to enable the
members of a group of autonomous agents to
asymptotically determine, in a decentralized manner,
the average of the initial values of their scalar
gossip variables. This paper discusses several
different deterministic protocols for gossiping which
avoid deadlocks and achieve consensus under different
assumptions. First considered is $T$-periodic gossiping
which is a gossiping protocol which stipulates that
each agent must gossip with the same neighbor exactly
once every $T$ time units. Among
the results discussed is the fact that if the
underlying graph characterizing neighbor relations is
a tree, convergence is exponential at a worst case
rate which is the same for all possible $T$ -periodic gossip
sequences associated with the graph. Many gossiping
protocols are request based which means simply that a
gossip between two agents will occur whenever one of
the two agents accepts a request to gossip placed by
the other. Three deterministic request-based
protocols are discussed. Each is guaranteed to not
deadlock and to always generate sequences of gossip
vectors which converge exponentially fast. It is
shown that worst case convergence rates can be
characterized in terms of the second largest singular
values of suitably defined doubly stochastic
matrices.},
doi = {10.1109/JPROC.2011.2159689},
}
@inproceedings{JL-ASM-BDOA-CY:11,
author = {Liu, J. and Morse, A. S. and Anderson, B. D. O. and
Yu, C.},
booktitle = {{IEEE} Conf.\ on Decision and Control and European
Control Conference},
pages = {1974-1979},
title = {Contractions for consensus processes},
year = {2011},
abstract = {Many distributed control algorithms of current
interest can be modeled by linear recursion equations
of the form x(t + 1) = M(t)x(t), t ≥ 1 where each
M(t) is a real-valued “stochastic” or “doubly
stochastic” matrix. Convergence of such recursions
often reduces to deciding when the sequence of matrix
productsM(1), M(2)M(1), M(3)M(2)M(1), … converges.
Certain types of stochastic and doubly stochastic
matrices have the property that any sequence of
products of such matrices of the form S1,
S2S1,
S3S2S1, …
converges exponentially fast. We explicitly
characterize the largest classes of stochastic and
doubly stochastic matrices with positive diagonal
entries which have these properties. The main goal of
this paper is to find a “semi-norm” with respect
to which matrices from these “convergability
classes” are contractions. For any doubly
stochastic matrix S such a semi-norm is identified
and is shown to coincide with the second largest
singular value of S.},
doi = {10.1109/CDC.2011.6160989},
}
@article{ZA-RF-AH-YC-TTG:20,
author = {Z. Askarzadeh and R. Fu and A. Halder and Y. Chen and
T. T. Georgiou},
journal = {IEEE Transactions on Automatic Control},
number = {2},
pages = {522-533},
title = {Stability theory of stochastic models in opinion
dynamics},
volume = {65},
year = {2020},
abstract = {We consider a certain class of nonlinear maps that
preserve the probability simplex, i.e., stochastic
maps, that are inspired by the DeGroot-Friedkin model
of belief/opinion propagation over influence
networks. The corresponding dynamical models describe
the evolution of the probability distribution of
interacting species. Such models where the
probability transition mechanism depends nonlinearly
on the current state are often referred to as
nonlinear Markov chains. In this paper we develop
stability results and study the behavior of
representative opinion models. The stability
certificates are based on the contractivity of the
nonlinear evolution in the l1-metric. We apply the
theory to two types of opinion models where the
adaptation of the transition probabilities to the
current state is exponential and linear,
respectively-both of these can display a wide range
of behaviors. We discuss continuous-time and other
generalizations.},
doi = {10.1109/TAC.2019.2912490},
}
@article{JJES:03,
author = {J.-J. E. Slotine},
journal = {International Journal of Adaptive Control and Signal
Processing},
number = {6},
pages = {397-416},
title = {Modular stability tools for distributed computation
and control},
volume = {17},
year = {2003},
doi = {10.1002/acs.754},
}
@incollection{JJES-WW:05b,
author = {J.-J. E. Slotine and W. Wang},
booktitle = {Cooperative Control. (Proceedings of the 2003 Block
Island Workshop on Cooperative Control)},
editor = {V. Kumar and N. E. Leonard and A. S. Morse},
pages = {207--228},
publisher = {Springer},
title = {A Study of Synchronization and Group Cooperation
Using Partial Contraction Theory},
year = {2005},
abstract = {Synchronization, collective behavior, and group
cooperation have been the object of extensive recent
research. A fundamental understanding of aggregate
motions in the natural world, such as bird flocks,
fish schools, animal herds, or bee swarms, for
instance, would greatly help in achieving desired
collective behaviors of artificial multi-agent
systems, such as vehicles with distributed
cooperative control rules. In [38], Reynolds
published his well-known computer model of ``boids,''
successfully forming an animation flock using three
local rules: collision avoidance, velocity matching,
and flock centering. Motivated by the growth of
colonies of bacteria, Viscek et al.[55] proposed a
similar discrete-time model which realizes heading
matching using information only from neighbors.
Viscek's model was later analyzed analytically [16,
52, 53]. Models in continuous-time [1, 22, 32, 33,
62] and combinations of Reynolds' three rules [21,
34, 35, 49, 50] were also studied. Related questions
can also be found e.g. in [3, 18, 20, 42], in
oscillator synchronization [48], as well as in
physics in the study of lasers [39] or of
Bose-Einstein condensation [17].},
doi = {10.1007/978-3-540-31595-7_12},
}
@article{SJC-JJES:09,
author = {S.-J. Chung and J.-J. E. Slotine},
journal = {IEEE Transactions on Robotics},
number = {3},
pages = {686-700},
title = {Cooperative Robot Control and Concurrent
Synchronization of {L}agrangian Systems},
volume = {25},
year = {2009},
abstract = {Concurrent synchronization is a regime where diverse
groups of fully synchronized dynamic systems stably
coexist. We study global exponential synchronization
and concurrent synchronization in the context of
Lagrangian systems control. In a network constructed
by adding diffusive couplings to robot manipulators
or mobile robots, a decentralized tracking control
law globally exponentially synchronizes an arbitrary
number of robots, and represents a generalization of
the average consensus problem. Exact nonlinear
stability guarantees and synchronization conditions
are derived by contraction analysis. The proposed
decentralized strategy is further extended to
adaptive synchronization and partial-state coupling.},
doi = {10.1109/TRO.2009.2014125},
}
@article{GR-JJES:10,
author = {G. Russo and J. J. E. Slotine},
journal = {Physical Review E},
number = {041919},
title = {Global convergence of quorum-sensing networks},
volume = {82},
year = {2010},
abstract = {In many natural synchronization phenomena,
communication between individual elements occurs not
di- rectly but rather through the environment. One of
these instances is bacterial quorum sensing, where
bacteria release signaling molecules in the
environment which in turn are sensed and used for
population coordination. Extending this motivation to
a general nonlinear dynamical system context, this
paper analyzes synchroniza- tion phenomena in
networks where communication and coupling between
nodes are mediated by shared dynamical quantities,
typically provided by the nodes’ environment. Our
model includes the case when the dynamics of the
shared variables themselves cannot be neglected or
indeed play a central part. Applications to examples
from system biology illustrate the approach.},
doi = {10.1103/PhysRevE.82.041919},
}
@article{LOC-LY:88,
author = {L. O. Chua and L. Yang},
journal = {IEEE Transactions on Circuits and Systems},
number = {10},
pages = {1257--1272},
title = {Cellular Neural Networks: Theory},
volume = {35},
year = {1988},
doi = {10.1109/31.7600},
}
@article{LS-RS:09,
author = {L. Scardovi and R. Sepulchre},
journal = {Automatica},
number = {11},
pages = {2557--2562},
title = {Synchronization in networks of identical linear
systems},
volume = {45},
year = {2009},
doi = {10.1016/j.automatica.2009.07.006},
}
@article{ES:88,
author = {Seneta, E.},
journal = {Advances in Applied Probability},
number = {1},
pages = {228-230},
title = {Perturbation of the stationary distribution measured
by ergodicity coefficients},
volume = {20},
year = {1988},
doi = {10.2307/1427277},
}
@article{GEC-CDM:01,
author = {G. E. Cho and C. D. Meyer},
journal = {Linear Algebra and its Applications},
number = {1},
pages = {137-150},
title = {Comparison of perturbation bounds for the stationary
distribution of a {M}arkov chain},
volume = {335},
year = {2001},
doi = {10.1016/S0024-3795(01)00320-2},
}
@article{SL:84,
author = {S. {\L}ojasiewicz},
journal = {Seminari di Geometria 1982-1983},
note = {Istituto di Geometria, Dipartimento di Matematica,
Universit{\`a} di Bologna, Italy},
pages = {115-117},
title = {Sur les trajectoires du gradient d'une fonction
analytique},
year = {1984},
}
@article{PAA-RM-BA:05,
author = {P.-A. Absil and R. Mahony and B. Andrews},
journal = {SIAM Journal on Control and Optimization},
number = {2},
pages = {531--547},
title = {Convergence of the Iterates of Descent Methods for
Analytic Cost Functions},
volume = {6},
year = {2005},
doi = {10.1137/040605266},
}
@article{JMD:66,
author = {J. M. Danskin},
journal = {SIAM Journal on Applied Mathematics},
number = {4},
pages = {641-664},
title = {The Theory of Max-Min, with Applications},
volume = {14},
year = {1966},
doi = {10.1137/0114053},
}
@book{JLL:1788,
address = {Paris},
author = {Joseph Louis Lagrange},
publisher = {Chez la Veuve Desaint},
title = {M\'ecanique Analytique},
year = {1788},
}
@article{JCM:1868,
author = {J. C. Maxwell},
journal = {Proceedings of the Royal Society. London. Series A.
Mathematical and Physical Sciences},
pages = {270-283},
title = {On Governors},
volume = {16},
year = {1868},
doi = {10.1098/rspl.1867.0055},
}
@book{WT-PGT:1867,
author = {W. Thomson and P. G. Tait},
publisher = {Oxford University Press},
title = {Treatise on Natural Philosophy},
year = {1867},
}
@book{AML:1892,
address = {Kharkov},
author = {Aleksandr Mikhailovich Lyapunov},
note = {Translation:~\citep{AML:1992}},
publisher = {Fakul\cprime{}teta i Khar\cprime{}kovskogo
Matematicheskogo Obshchestva},
title = {Ob\v{s}\v{c}aya zada\v{c}a ob usto\u{\i}\v{c}ivosti
dvi\v{z}eniya},
year = {1892},
}
@article{EAB-NNK:52,
author = {E. A. Barbashin and N. N. Krasovski\u{\i}},
journal = {Doklady Akademii Nauk SSSR},
note = {(In Russian)},
number = {3},
pages = {453-456},
title = {On Global Stability of Motion},
volume = {86},
year = {1952},
}
@article{JPL:60,
author = {J. P. LaSalle},
journal = {IRE Transactions on Circuit Theory},
pages = {520-527},
title = {Some extensions of {L}iapunov's second method},
volume = {CT-7},
year = {1960},
doi = {10.1109/TCT.1960.1086720},
}
@article{JPL:68,
author = {J. P. LaSalle},
journal = {Journal of Differential Equations},
pages = {57--65},
title = {Stability Theory for Ordinary Differential Equations},
volume = {4},
year = {1968},
doi = {10.1016/0022-0396(68)90048-X},
}
@book{JPL:76,
author = {J. P. LaSalle},
publisher = {SIAM},
title = {The Stability of Dynamical Systems},
year = {1976},
doi = {10.1137/1.9781611970432},
isbn = {9780898710229},
}
@book{NGC:61,
author = {Nikolai Gurevich Chetaev},
note = {Translation from Russian by M.~Nadler},
publisher = {Pergamon},
title = {The Stability of Motion},
year = {1961},
}
@book{WH:67,
author = {W. Hahn},
publisher = {Springer},
title = {Stability of Motion},
year = {1967},
isbn = {978-3-642-50085-5},
}
@book{EDS:98,
author = {E. D. Sontag},
edition = {2},
publisher = {Springer},
title = {Mathematical Control Theory: Deterministic Finite
Dimensional Systems},
year = {1998},
isbn = {0387984895},
}
@book{HKK:02,
author = {H. K. Khalil},
edition = {3},
publisher = {Prentice Hall},
title = {Nonlinear Systems},
year = {2002},
isbn = {0130673897},
}
@book{WMH-SS:74,
author = {M. W. Hirsch and S. Smale},
publisher = {Academic Press},
title = {Differential Equations, Dynamical Systems and Linear
Algebra},
year = {1974},
isbn = {0123495504},
}
@book{VIA:92,
author = {Vladimir I. Arnol'd},
note = {Translation of the third Russian edition by R.~Cooke},
publisher = {Springer},
title = {Ordinary Differential Equations},
year = {1992},
isbn = {3-540-54813-0},
}
@book{JG-PH:90,
author = {J. Guckenheimer and P. Holmes},
publisher = {Springer},
title = {Nonlinear Oscillations, Dynamical Systems, and
Bifurcations of Vector Fields},
year = {1990},
isbn = {0387908196},
}
@book{WMH-VC:08,
author = {W. M. Haddad and V. Chellaboina},
publisher = {Princeton University Press},
title = {Nonlinear Dynamical Systems and Control: A
Lyapunov-Based Approach},
year = {2008},
isbn = {9780691133294},
}
@book{RG-RGS-ART:12,
author = {R. Goebel and R. G. Sanfelice and A. R. Teel},
publisher = {Princeton University Press},
title = {Hybrid Dynamical Systems: Modeling, Stability, and
Robustness},
year = {2012},
isbn = {9780691153896},
}
@book{FHC-YSL-RJS-PRW:98,
author = {F. H. Clarke and Y.S. Ledyaev and R. J. Stern and
P. R. Wolenski},
publisher = {Springer},
title = {Nonsmooth Analysis and Control Theory},
year = {1998},
isbn = {0387983368},
}
@article{JC:08-csm,
author = {J. Cort{\'e}s},
journal = {{IEEE} Control Systems},
number = {3},
pages = {36-73},
title = {Discontinuous dynamical systems},
volume = {28},
year = {2008},
abstract = {This paper considers discontinuous dynamical systems,
i.e., systems whose associated vector field is a
discontinuous function of the state. Discontinuous
dynamical systems arise in a large number of
applications, including optimal control, nonsmooth
mechanics, and robotic manipulation. Independently of
the particular application, one always faces similar
questions when dealing with discontinuous dynamical
systems. The most basic one is the notion of
solution. We begin by introducing the notions of
Caratheodory, Filippov and sample-and-hold solutions,
discuss existence and uniqueness results for them,
and examine various examples. We also give specific
pointers to other notions of solution defined in the
literature. Once the notion of solution has been
settled, we turn our attention to the analysis of
stability of discontinuous systems. We introduce the
concepts of generalized gradient of locally Lipschitz
functions and proximal subdifferential of lower
semicontinuous functions. Building on these notions,
we establish monotonic properties of candidate
Lyapunov functions along the solutions. These results
are key in providing suitable generalizations of
Lyapunov stability theorems and the LaSalle
Invariance Principle. We illustrate the applicability
of these results in a class of nonsmooth gradient
flows.},
doi = {10.1109/MCS.2008.919306},
}
@article{ZL-BF-MM:07,
author = {Z. Lin and B. Francis and M. Maggiore},
journal = {SIAM Journal on Control and Optimization},
number = {1},
pages = {288-307},
title = {State agreement for continuous-time coupled nonlinear
systems},
volume = {46},
year = {2007},
abstract = {Two related problems are treated in continuous time.
First, the state agreement problem is studied for
coupled nonlinear differential equations. The vector
fields can switch within a finite family. Associated
to each vector field is a directed graph based in a
natural way on the interaction structure of the
subsystems. Generalizing the work of Moreau, under
the assumption that the vector fields satisfy a
certain subtangentiality condition, it is proved that
asymptotic state agreement is achieved if and only if
the dynamic interaction digraph has the property of
being sufficiently connected over time. The proof
uses nonsmooth analysis. Second, the rendezvous
problem for kinematic point-mass mobile robots is
studied when the robotsÃ fields of view have a
fixed radius. The circumcenter control law of Ando et
al. [IEEE Trans. Robotics Automation, 15 (1999), pp.
818Ã 828] is shown to solve the problem. The
rendezvous problem is a kind of state agreement
problem, but the interaction structure is state
dependent.},
doi = {10.1137/050626405},
}
@book{WR-RWB:08,
author = {W. Ren and R. W. Beard},
publisher = {Springer},
series = {Communications and Control Engineering},
title = {Distributed Consensus in Multi-vehicle Cooperative
Control},
year = {2008},
isbn = {978-1-84800-014-8},
}
@book{FB-JC-SM:09,
author = {F. Bullo and J. Cort{\'e}s and S. Mart{\'\i}nez},
publisher = {Princeton University Press},
title = {Distributed Control of Robotic Networks},
year = {2009},
isbn = {978-0-691-14195-4},
url = {http://www.coordinationbook.info},
}
@book{MM-ME:10,
author = {M. Mesbahi and M. Egerstedt},
publisher = {Princeton University Press},
title = {Graph Theoretic Methods in Multiagent Networks},
year = {2010},
isbn = {9781400835355},
}
@book{HB-MA-JW:11,
author = {H. Bai and M. Arcak and J. Wen},
publisher = {Springer},
title = {Cooperative Control Design},
year = {2011},
isbn = {1461429072},
}
@book{EC-BP-AT:14,
author = {E. Cristiani and B. Piccoli and A. Tosin},
publisher = {Springer},
title = {Multiscale Modeling of Pedestrian Dynamics},
year = {2014},
isbn = {978-3-319-06619-6},
}
@book{BAF-MM:16,
author = {B. A. Francis and M. Maggiore},
publisher = {Springer},
title = {Flocking and Rendezvous in Distributed Robotics},
year = {2016},
isbn = {978-3-319-24727-4},
}
@book{MA-CM-AP:16,
author = {M. Arcak and C. Meissen and A. Packard},
publisher = {Springer},
title = {Networks of Dissipative Systems: Compositional
Certification of Stability, Performance, and Safety},
year = {2016},
doi = {10.1007/978-3-319-29928-0},
isbn = {978-3-319-29928-0},
}
@article{SM-JC-FB:04n,
author = {S. Mart{\'\i}nez and J. Cort{\'e}s and F. Bullo},
journal = {{IEEE} Control Systems},
number = {4},
pages = {75-88},
title = {Motion Coordination with Distributed Information},
volume = {27},
year = {2007},
abstract = {This paper surveys recently-developed theoretical
tools for the analysis and design of coordination
algorithms for networks of mobile autonomous agents.
First, various motion coordination tasks are encoded
into aggregate cost functions from Geometric
Optimization. Second, the limited communication
capabilities of the mobile agents are modeled via the
notions of proximity graphs from Computational
Geometry and of spatially distributed maps. Finally,
we illustrate how to apply these tools to design and
analyze scalable cooperative strategies in a variety
of motion coordination problems such as deployment,
rendezvous, and flocking.},
doi = {10.1109/MCS.2007.384124},
}
@article{WR-RWB-EMA:07,
author = {W. Ren and R. W. Beard and E. M. Atkins},
journal = {{IEEE} Control Systems},
number = {2},
pages = {71-82},
title = {Information consensus in multivehicle cooperative
control},
volume = {27},
year = {2007},
doi = {10.1109/MCS.2007.338264},
}
@incollection{FG-LS:10,
author = {F. Garin and L. Schenato},
booktitle = {Networked Control Systems},
editor = {A. Bemporad and M. Heemels and M. Johansson},
pages = {75-107},
publisher = {Springer},
title = {A Survey on Distributed Estimation and Control
Applications Using Linear Consensus Algorithms},
year = {2010},
doi = {10.1007/978-0-85729-033-5_3},
}
@article{YC-WY-WR-GC:13,
author = {Y. Cao and W. Yu and W. Ren and G. Chen},
journal = {IEEE Transactions on Industrial informatics},
number = {1},
pages = {427--438},
title = {An overview of recent progress in the study of
distributed multi-agent coordination},
volume = {9},
year = {2013},
doi = {10.1109/TII.2012.2219061},
}
@article{KKO-MCP-HSA:15,
author = {K.-K. Oh and M.-C. Park and H.-S. Ahn},
journal = {Automatica},
pages = {424--440},
title = {A survey of multi-agent formation control},
volume = {53},
year = {2015},
doi = {10.1016/j.automatica.2014.10.022},
}
@book{FRG:59ab,
address = {New York},
author = {Felix R. Gantmacher},
note = {Translation of German edition by K.~A.~Hirsch},
publisher = {Chelsea},
title = {The Theory of Matrices},
volume = {1 and 2},
year = {1959},
isbn = {0-8218-1376-5 and 0-8218-2664-6},
}
@article{JRPF:56,
author = {J. R. P. {French~Jr.}},
journal = {Psychological Review},
number = {3},
pages = {181--194},
title = {A formal theory of social power},
volume = {63},
year = {1956},
doi = {10.1037/h0046123},
}
@incollection{FH:59,
author = {F. Harary},
booktitle = {Studies in Social Power},
editor = {D. Cartwright},
pages = {168--182},
publisher = {University of Michigan},
title = {A criterion for unanimity in {F}rench's theory of
social power},
year = {1959},
isbn = {0879442301},
url = {http://psycnet.apa.org/psycinfo/1960-06701-006},
}
@incollection{RPA:64,
author = {R. P. Abelson},
booktitle = {Contributions to Mathematical Psychology},
editor = {N. Frederiksen and H. Gulliksen},
pages = {142--160},
publisher = {Holt, Rinehart, \& Winston},
title = {Mathematical models of the distribution of attitudes
under controversy},
volume = {14},
year = {1964},
isbn = {0030430100},
}
@article{MHDG:74,
author = {M. H. DeGroot},
journal = {Journal of the American Statistical Association},
number = {345},
pages = {118-121},
title = {Reaching a Consensus},
volume = {69},
year = {1974},
abstract = {Consider a group of individuals who must act together
as a team or committee, and suppose that each
individual in the group has his own subjective
probability distribution for the unknown value of
some parameter. A model is presented which describes
how the group might reach agreement on a common
subjective probability distribution for the parameter
by pooling their individual opinions. The process
leading to the consensus is explicitly described and
the common distribution that is reached is explicitly
determined. The model can also be applied to problems
of reaching a consensus when the opinion of each
member of the group is represented simply as a point
estimate of the parameter rather than as a
probability distribution.},
doi = {10.1080/01621459.1974.10480137},
}
@article{MF:73,
author = {M. Fiedler},
journal = {Czechoslovak Mathematical Journal},
number = {2},
pages = {298--305},
publisher = {Institute of Mathematics, Academy of Sciences of the
Czech Republic},
title = {Algebraic connectivity of graphs},
volume = {23},
year = {1973},
url = {http://dml.cz/dmlcz/101168},
}
@article{ROS-RMM:03c,
author = {R. Olfati-Saber and R. M. Murray},
journal = {IEEE Transactions on Automatic Control},
number = {9},
pages = {1520-1533},
title = {Consensus problems in networks of agents with
switching topology and time-delays},
volume = {49},
year = {2004},
doi = {10.1109/TAC.2004.834113},
}
@article{JNT-DPB-MA:86,
author = {J. N. Tsitsiklis and D. P. Bertsekas and M. Athans},
journal = {IEEE Transactions on Automatic Control},
number = {9},
pages = {803-812},
title = {Distributed asynchronous deterministic and stochastic
gradient optimization algorithms},
volume = {31},
year = {1986},
abstract = {Asynchronous distributed iterative optimization
algorithms are modeled for the following cases in
which each processor does not need to communicate to
each other processor at each time instance:
processors may keep performing computations without
having to wait until they receive the messages that
have been transmitted to them; processors are allowed
to remain idle some of the time; some processors may
perform computations faster than others. A model for
asynchronous distributed computation is presented and
then the convergence of natural asynchronous
distributed versions of a large class of
deterministic and stochastic gradient-like algorithms
is analyzed. It is shown that such algorithms retain
the desirable convergence properties of their
centralized counterparts, provided that the time
between consecutive communications between processors
and communication delays is not too large. (19
References).},
doi = {10.1109/TAC.1986.1104412},
}
@article{AJ-JL-ASM:02,
author = {A. Jadbabaie and J. Lin and A. S. Morse},
journal = {IEEE Transactions on Automatic Control},
number = {6},
pages = {988-1001},
title = {Coordination of groups of mobile autonomous agents
using nearest neighbor rules},
volume = {48},
year = {2003},
abstract = {In a recent Physical Review Letters article, Vicsek
et al. propose a simple but compelling discrete-time
model of n autonomous agents (i.e., points or
particles) all moving in the plane with the same
speed but with different headings. Each agent's
heading is updated using a local rule based on the
average of its own heading plus the headings of its
"neighbors." In their paper, Vicsek et al. provide
simulation results which demonstrate that the nearest
neighbor rule they are studying can cause all agents
to eventually move in the same direction despite the
absence of centralized coordination and despite the
fact that each agent's set of nearest neighbors
change with time as the system evolves. This paper
provides a theoretical explanation for this observed
behavior. In addition, convergence results are
derived for several other similarly inspired models.
The Vicsek model proves to be a graphic example of a
switched linear system which is stable, but for which
there does not exist a common quadratic Lyapunov
function.},
doi = {10.1109/TAC.2003.812781},
}
@article{PvM-JO-RK:09,
author = {P. {Van~Mieghem} and J. Omic and R. Kooij},
journal = {IEEE/ACM Transactions on Networking},
number = {1},
pages = {1-14},
title = {Virus spread in networks},
volume = {17},
year = {2009},
doi = {10.1109/TNET.2008.925623},
}
@book{HWH-JAY:84,
author = {H. W. Hethcote and J. A. Yorke},
publisher = {Springer},
title = {Gonorrhea Transmission Dynamics and Control},
year = {1984},
doi = {10.1007/978-3-662-07544-9},
isbn = {978-3-540-13870-9},
}
@article{SB-LP:98,
author = {S. Brin and L. Page},
journal = {Computer Networks},
pages = {107-117},
title = {The anatomy of a large-scale hypertextual {W}eb
search engine},
volume = {30},
year = {1998},
doi = {10.1016/S0169-7552(98)00110-X},
}
@article{NEF:91,
author = {N. E. Friedkin},
journal = {American Journal of Sociology},
number = {6},
pages = {1478-1504},
title = {Theoretical foundations for centrality measures},
volume = {96},
year = {1991},
doi = {10.1086/229694},
}
@article{DF:72,
author = {D. Fife},
journal = {Mathematical Biosciences},
number = {3},
pages = {311--315},
title = {Which linear compartmental systems contain traps?},
volume = {14},
year = {1972},
doi = {10.1016/0025-5564(72)90082-X},
}
@article{DMF-JAJ:75,
author = {D. M. Foster and J. A. Jacquez},
journal = {Mathematical Biosciences},
number = {1},
pages = {89--97},
title = {Multiple zeros for eigenvalues and the multiplicity
of traps of a linear compartmental system},
volume = {26},
year = {1975},
doi = {10.1016/0025-5564(75)90096-6},
}
@article{RPA-PYC:00,
author = {R. P. Agaev and P. Y. Chebotarev},
journal = {Automation and Remote Control},
number = {9},
pages = {1424-1450},
title = {The matrix of maximum out forests of a digraph and
its applications},
volume = {61},
year = {2000},
abstract = {We study the maximum out forests of a (weighted)
digraph and the matrix of maximum out forests. A
maximum out forest of a digraph Gamma is a spanning
subgraph of Gamma that consists of disjoint diverging
trees and has the maximum possible number of arcs. If
a digraph contains out arborescences, then maximum
out forests coincide with them. We consider Markov
chains related to a weighted digraph and prove that
the matrix of Cesaro limiting probabilities of such a
chain coincides with the normalized matrix of maximum
out forests. This provides an interpretation for the
matrix of Cesasro limiting probabilities of an
arbitrary stationary finite Markov chain in terms of
the weight of maximum out forests. We discuss the
applications of the matrix of maximum out forests and
its transposition, the matrix of limiting
accessibilities of a digraph, to the problems of
preference aggregation, measuring the vertex
proximity, and uncovering the structure of a
digraph.},
url = {https://arxiv.org/pdf/math/0602059},
}
@article{ZL-BF-MM:05,
author = {Z. Lin and B. Francis and M. Maggiore},
journal = {IEEE Transactions on Automatic Control},
number = {1},
pages = {121-127},
title = {Necessary and sufficient graphical conditions for
formation control of unicycles},
volume = {50},
year = {2005},
doi = {10.1109/TAC.2004.841121},
}
@article{WR-RWB:05,
author = {W. Ren and R. W. Beard},
journal = {IEEE Transactions on Automatic Control},
number = {5},
pages = {655-661},
title = {Consensus seeking in multiagent systems under
dynamically changing interaction topologies},
volume = {50},
year = {2005},
abstract = {This note considers the problem of information
consensus among multiple agents in the presence of
limited and unreliable information exchange with
dynamically changing interaction topologies. Both
discrete and continuous update schemes are proposed
for information consensus. This note shows that
information consensus under dynamically changing
interaction topologies can be achieved asymptotically
if the union of the directed interaction graphs have
a spanning tree frequently enough as the system
evolves.},
doi = {10.1109/TAC.2005.846556},
}
@book{AML:1992,
author = {Aleksandr Mikhailovich Lyapunov},
note = {Translation from Russian by A.~T.~Fuller},
publisher = {Taylor \& Francis},
title = {The General Problem of the Stability of Motion},
year = {1992},
}