@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 robotsÕ fields of view have a fixed radius. The circumcenter control law of Ando et al. [IEEE Trans. Robotics Automation, 15 (1999), pp. 818Ð 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}, }