Asymptotic Stability and Smooth Lyapunov Functions

Clarke, Frank H.; Ledyaev, Yu. S.; Stern, Ronald J.

doi:10.1006/jdeq.1998.3476

Cited by 253 publications

(228 citation statements)

References 25 publications

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“…the variables V xi and C ν,i have values such that the constraints (4), (5), and (6) are all fulfilled, then it is always possible to algorithmically find a feasible solution, e.g. by the simplex algorithm.…”

Section: The Linear Programming Problemmentioning

confidence: 99%

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Revised CPA method to compute Lyapunov functions for nonlinear systems

Giesl

Hafstein

2014

Journal of Mathematical Analysis and Applications

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The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov function for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium's basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C 2 and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.

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Section: The Linear Programming Problemmentioning

confidence: 99%

“…They can be used for very general systems, e.g. nonautonomous systems [22,35,16], arbitrary switched nonautonomous systems [15], or differential inclusions [5], but in this paper we concentrate on autonomous systems.…”

Section: Introductionmentioning

confidence: 99%

Revised CPA method to compute Lyapunov functions for nonlinear systems

Giesl

Hafstein

2014

Journal of Mathematical Analysis and Applications

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“…is strongly asymptotically stable if every trajectory x(t) is defined for all t ≥ 0 and satisfies lim t→+∞ x(t) = 0, and if in addition the origin has the familiar local property known as 'Lyapunov stability'. The following result, which unifies and extends several classical theorems dealing with the uncontrolled case, is due to Clarke, Ledyaev and Stern [9]: Theorem 1. Let F have compact convex nonempty values and closed graph.…”

Section: Strong Stabilitymentioning

confidence: 56%

Lyapunov Functions and Feedback in Nonlinear Control

Clarke

2004

Optimal Control, Stabilization and Nonsmooth Analysis

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Summary. The method of Lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems. For nonlinear systems, it turns out to be essential to consider nonsmooth Lyapunov functions, even if the underlying control dynamics are themselves smooth. We synthesize in this article a number of recent developments bearing upon the regularity properties of Lyapunov functions. A novel feature of our approach is that the guidability and stability issues are decoupled. For each of these issues, we identify various regularity classes of Lyapunov functions and the system properties to which they correspond. We show how such regularity properties are relevant to the construction of stabilizing feedbacks. Such feedbacks, which must be discontinuous in general, are implemented in the sample-and-hold sense. We discuss the equivalence between open-loop controllability, feedback stabilizability, and the existence of Lyapunov functions with appropriate regularity properties. The extent of the equivalence confirms the cogency of the new approach summarized here.

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“…Thus x λ is globally asymptotically stable. Thanks to the basic theory of dynamical systems (see [1,7,17] etc. ), we know that x λ attracts each bounded subset of ‫ޒ‬ 1 .…”

Section: Theorem 27 In Addition To the Hypotheses In Theorem 23 Amentioning

confidence: 99%

Equi-Attraction and the Continuous Dependence of Attractors on Parameters

Li¹,

Kloeden²

2004

Glasgow Math. J.

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Abstract. The equi-attraction properties concerning the global attractors A λ of dynamical systems S λ (t) with parameter λ ∈ , where is a compact metric space, are investigated. In particular, under appropriate conditions, it is shown that the equiattraction of the family {A λ } is equivalent to the continuity of A λ in λ with respect to the Hausdorff distance.2000 Mathematics Subject Classification. Primary 47J25, 37L30, 34C20.

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Asymptotic Stability and Smooth Lyapunov Functions

Cited by 253 publications

References 25 publications

Revised CPA method to compute Lyapunov functions for nonlinear systems

Revised CPA method to compute Lyapunov functions for nonlinear systems

Lyapunov Functions and Feedback in Nonlinear Control

Equi-Attraction and the Continuous Dependence of Attractors on Parameters

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