A Smooth Lyapunov Function from a Class-kl Estimate Involving Two Positive Semideenite Functions

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 if 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. AMS Subject Classification. 34A60, 34D20, 34B25. Received November 3, 1999. Revised May 24, 2000. Basic definitions • Given a set A, A stands for the closure of A, Å stands for the interior set of A, coA stands for the closed convex hull of A and ∂A stands for the boundary of A. • The notation x → ∂A∞ indicates a sequence of points x belonging to A converging to a point on the boundary of A or, if A is unbounded, having the property |x| → ∞. • Given a closed set A ⊂ R and a point x ∈ R, |x|A denotes the distance from x to A. • A function α : R≥0 → R≥0 is said to belong to class-K (α ∈ K) if it is continuous, zero at zero, and strictly increasing. It is said to belong to class-K∞ if, in addition, it is unbounded. • A function β : R≥0 × R≥0 → R≥0 is said to belong to class-KL if, for each t ≥ 0, β(·, t) is nondecreasing and lims→0+ β(s, t) = 0, and, for each s ≥ 0, β(s, ·) is nonincreasing and limt→∞ β(s, t) = 0. The requirements imposed for a function to be of class-KL are slightly weaker than usual. In particular, β(·, t) is not required to be continuous or strictly increasing. See, also, Remark 3.

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