A Contractive Approach to Separable Lyapunov Functions for Monotone Systems

Monotone systems preserve a partial ordering of states along system trajectories and are often amenable to separable Lyapunov functions that are either the sum or the maximum of a collection of functions of a scalar argument. In this paper, we consider constructing separable Lyapunov functions for monotone systems that are also contractive, that is, the distance between any pair of trajectories exponentially decreases. The distance is defined in terms of a possibly state-dependent norm. When this norm is a weighted one-norm, we obtain conditions which lead to sum-separable Lyapunov functions, and when this norm is a weighted infinity-norm, symmetric conditions lead to max-separable Lyapunov functions. In addition, we consider two classes of Lyapunov functions: the first class is separable along the system's state, and the second class is separable along components of the system's vector field. The latter case is advantageous for many practically motivated systems for which it is difficult to measure the system's state but easier to measure the system's velocity or rate of change. In addition, we present an algorithm based on sum-of-squares programming to compute such separable Lyapunov functions. We provide several examples to demonstrate our results.Comment: arXiv admin note: text overlap with arXiv:1609.0625

Diffusion with nonlocal Dirichlet boundary conditions on unbounded domains

We consider a second order differential operator $\mathscr{A}$ on an (typically unbounded) open and Dirichlet regular set $\Omega\subset \mathbb{R}^d$ and subject to nonlocal Dirichlet boundary conditions of the form u(z) = \int_\Omega u(x)\mu (z, dx) \quad \mbox{ for } z\in \partial \Omega. Here, $\mu : \partial\Omega \to \mathscr{M}(\Omega)$ is a $\sigma (\mathscr{M}(\Omega), C_b(\Omega))$-continuous map taking values in the probability measures on $\Omega$. Under suitable assumptions on the coefficients in $\mathscr{A}$, which may be unbounded, we prove that a realization $A_\mu$ of $\mathscr{A}$ subject to the nonlocal boundary condition, generates a (not strongly continuous) semigroup on $L^\infty(\Omega)$. We also establish a sufficient condition for this semigroup to be Markovian and prove that in this case, it enjoys the strong Feller property. We also study the asymptotic behavior of the semigroup.Comment: 27 pages, no figures. This is a revision based on the comments of the referee

Stability Theory of Stochastic Models in Opinion Dynamics

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 {\em 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 $\ell_1$-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.Comment: 11 pages, 6 figure

Stability Analysis of Monotone Systems via Max-separable Lyapunov Functions

We analyze stability properties of monotone nonlinear systems via max-separable Lyapunov functions, motivated by the following observations: first, recent results have shown that asymptotic stability of a monotone nonlinear system implies the existence of a max-separable Lyapunov function on a compact set; second, for monotone linear systems, asymptotic stability implies the stronger properties of D-stability and insensitivity to time-delays. This paper establishes that for monotone nonlinear systems, equivalence holds between asymptotic stability, the existence of a max-separable Lyapunov function, D-stability, and insensitivity to bounded and unbounded time-varying delays. In particular, a new and general notion of D-stability for monotone nonlinear systems is discussed and a set of necessary and sufficient conditions for delay-independent stability are derived. Examples show how the results extend the state-of-the-art

Distributed Nonlinear Control Design using Separable Control Contraction Metrics

This paper gives convex conditions for synthesis of a distributed control system for large-scale networked nonlinear dynamic systems. It is shown that the technique of control contraction metrics (CCMs) can be extended to this problem by utilizing separable metric structures, resulting in controllers that only depend on information from local sensors and communications from immediate neighbours. The conditions given are pointwise linear matrix inequalities, and are necessary and sufficient for linear positive systems and certain monotone nonlinear systems. Distributed synthesis methods for systems on chordal graphs are also proposed based on SDP decompositions. The results are illustrated on a problem of vehicle platooning with heterogeneous vehicles, and a network of nonlinear dynamic systems with over 1000 states that is not feedback linearizable and has an uncontrollable linearizationComment: Conditionally accepted to IEEE Transactions on Control of Networked System

Non-Euclidean Contraction Theory for Monotone and Positive Systems

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 semi-inner products, we characterize strongly contracting monotone systems in non-Euclidean spaces. We apply this framework to i) study stability of monotone separable systems, ii) establish strong contractivity of excitatory Hopfield neural networks, and iii) obtain a novel version of the Matrosov-Bellman comparison lemma. We also characterize equilibrium-contracting positive systems in non-Euclidean spaces and provide a sufficient condition for equilibrium-contractivity using conic measures. We apply this framework to i) study contractivity of positive separable systems, and ii) develop a novel comparison-based framework for studying interconnected systems

Performance guarantees for model-based Approximate Dynamic Programming in continuous spaces

We study both the value function and Q-function formulation of the Linear Programming approach to Approximate Dynamic Programming. The approach is model-based and optimizes over a restricted function space to approximate the value function or Q-function. Working in the discrete time, continuous space setting, we provide guarantees for the fitting error and online performance of the policy. In particular, the online performance guarantee is obtained by analyzing an iterated version of the greedy policy, and the fitting error guarantee by analyzing an iterated version of the Bellman inequality. These guarantees complement the existing bounds that appear in the literature. The Q-function formulation offers benefits, for example, in decentralized controller design, however it can lead to computationally demanding optimization problems. To alleviate this drawback, we provide a condition that simplifies the formulation, resulting in improved computational times.Comment: 18 pages, 5 figures, journal pape

Graph-Theoretic Stability Conditions for Metzler Matrices and Monotone Systems

This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from 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 derive two sets of stability conditions based on two forms of input-to-state stability gains for Metzler systems, namely max-interconnection gains and sum-interconnection gains. 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

Payoff Dynamics Model and Evolutionary Dynamics Model: Feedback and Convergence to Equilibria

This tutorial article puts forth a framework to analyze the noncooperative strategic interactions among the members of a large population of bounded rationality agents. Our approach hinges on, unifies and generalizes existing methods and models predicated in evolutionary and population games. It does so by adopting a system-theoretic formalism that is well-suited for a broad engineering audience familiar with the basic tenets of nonlinear dynamical systems, Lyapunov stability, storage functions, and passivity. The framework is pertinent for engineering applications in which a large number of agents have the authority to select and repeatedly revise their strategies. A mechanism that is inherent to the problem at hand or is designed and implemented by a coordinator ascribes a payoff to each possible strategy. Typically, the agents will prioritize switching to strategies whose payoff is either higher than the current one or exceeds the population average. The article puts forth a systematic methodology to characterize the stability of the dynamical system that results from the feedback interaction between the payoff mechanism and the revision process. This is important because the set of stable equilibria is an accurate predictor of the population's long-term behavior. The article includes rigorous proofs and examples of application of the stability results, which also extend the state of the art because, unlike previously published work, they allow for a rather general class of dynamical payoff mechanisms. The new results and concepts proposed here are thoroughly compared to previous work, methods and applications of evolutionary and population games

Subadditive and Multiplicative Ergodic Theorems

A result for subadditive ergodic cocycles is proved that provides more delicate information than Kingman's subadditive ergodic theorem. As an application we deduce a multiplicative ergodic theorem generalizing an earlier result of Karlsson-Ledrappier, showing that the growth of a random product of semi-contractions is always directed by some horofunction. We discuss applications of this result to ergodic cocycles of bounded linear operators, holomorphic maps and topical operators, as well as a random mean ergodic theorem.Comment: 20 page