Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Holder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.

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Finite-Time Stability of Continuous Autonomous Systems

We study a simple but compelling model of network of agents interacting via time-dependent communication links. The model finds application in a variety of fields including synchronization, swarming and distributed decision making. In the model, each agent updates his current state based upon the current information received from neighboring agents. Necessary and/or sufficient conditions for the convergence of the individual agents' states to a common value are presented, thereby extending recent results reported in the literature. The stability analysis is based upon a blend of graph-theoretic and system-theoretic tools with the notion of convexity playing a central role. The analysis is integrated within a formal framework of set-valued Lyapunov theory, which may be of independent interest. Among others, it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper.

Stability of multiagent systems with time-dependent communication links

Discrete dynamical systems: Limit sets Stability of invariant sets and asymptotically smooth maps Examples of asymptotically smooth maps Dissipativeness and global attractors Dependence on parameters Fixed point theorems Stability relative to the global attractor and Morse-Smale maps Dimension of the global attractor Dissipativeness in two spaces Continuous dynamical systems: Limit sets Asymptotically smooth and $\alpha$-contracting semigroups Stability of invariant sets Dissipativeness and global attractors Dependence on parameters Periodic processes Skew product flows Gradient flows Dissipativeness in two spaces Properties of the flow on the attractor Applications: Retarded functional differential equations Sectorial evolutionary equations A scalar parabolic equation The Navier-Stokes equation Neutral functional differential equations Some abstract evolutionary equations A one-dimensional damped wave equation A three-dimensional damped wave equation Remarks on other applications Dependence on parameters and approximation of the attractor.

Asymptotic Behavior of Dissipative Systems

Systems that are competitive and possess symmetric interactions admit a global Lyapunov function. However, a global Lyapunov function whose equilibrium set can be effectively analyzed has not yet been discovered. It remains an open question whether the Lyapunov function approach, which requires a study of equilibrium points, or an alternative global approach, such as the Lyapunov functional approach, which sidesteps a direct study of equilibrium points will ultimately handle all of the physically important cases.

Absolute stability of global pattern formation and parallel memory storage by competitive neural networks

Robust stability and control for systems that combine continuous-time and discrete-time dynamics. This article is a tutorial on modeling the dynamics of hybrid systems, on the elements of stability theory for hybrid systems, and on the basics of hybrid control. The presentation and selection of material is oriented toward the analysis of asymptotic stability in hybrid systems and the design of stabilizing hybrid controllers. Our emphasis on the robustness of asymptotic stability to data perturbation, external disturbances, and measurement error distinguishes the approach taken here from other approaches to hybrid systems. While we make some connections to alternative approaches, this article does not aspire to be a survey of the hybrid system literature, which is vast and multifaceted.

Hybrid dynamical systems

Stability theory by Liapunov's second method

I. Preliminaries.- 1. Liapunov Functions.- 2. Almost Periodic Functions.- 3. Asymptotically Almost Periodic Functions.- 4. Quasi-Periodic Functions.- 5. Boundary Value Problem.- II. Stability and Boundedness.- 6. Stability of a Solution.- 7. Asymptotic Stability of a Solution.- 8. Boundedness of Solutions.- 9. Asymptotic Stability in the Large.- 10. Asymptotic Behavior of Solutions.- 11. Converse Theorems.- 12. Total Stability.- 13. Inherited Properties in Almost Periodic Systems.- 14. Uniformly Asymptotic Stability in Almost Periodic Systems.- III. Existence Theorems for Periodic Solutions and Almost Periodic Solutions.- 15. Existence Theorems for Periodic Solutions.- 16. Existence Theorems for Almost Periodic Solutions.- 17. Separation Condition in Almost Periodic Systems.- 18. Uniform Stability and Existence of Almost Periodic Solutions.- 19. Existence of Almost Periodic Solutions by Liapunov Functions.- References.

太郎吉沢

Papers

Stability theory by Liapunov's second method

Stability theory and the existence of periodic solutions and almost periodic solutions

An Invariance Principle in the Theory of Stability (常微分方程式及び函数微分方程式研究会報告集)

Oscillatory Property of Solutions of Second Order Differential Equations (常微分方程式と非線形力学)

太郎 吉沢

Papers

Stability theory by Liapunov's second method

Stability theory and the existence of periodic solutions and almost periodic solutions

An Invariance Principle in the Theory of Stability (常微分方程式及び函数微分方程式研究会報告集)

Oscillatory Property of Solutions of Second Order Differential Equations (常微分方程式と非線形力学)

太郎吉沢