A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions

被引：0

作者：

Teel, Andrew R. ^{[1
]}

Praly, Laurent ^{[2
]}

机构：

[1] ECE Department, University of California, Santa Barbara, CA 93106, United States

[2] Centre Automatique et Systèmes, École des Mines de Paris, 35 rue Saint Honoré, 77305 Fontainebleau Cedex, France

来源：

| 2000年 / EDP Sciences卷

关键词：

Approximation theory - Asymptotic stability - Boundary conditions - Conformal mapping - Convergence of numerical methods - Functions - Ordinary differential equations - Perturbation techniques - Robustness (control systems) - Set theory - State estimation;

D O I：

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摘要：

We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class-KL estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class-KL estimate, exists if and only the class-KL estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether all class-KL estimates are robust. One sufficient condition for robustness is that the original differential inclusion is locally Lipschitz. Another sufficient condition is that the two positive semidefinite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for differential equations and differential inclusions that have appeared in the literature.

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