Nonlocal functions of Lyapunov-Krasovskii type. (Russian) Zbl 0623.34036

Consider the differential equation (*) \(x'=P(x,t)\), where \(P: \Omega\to \mathbb R^ n\), \(\Omega \subset \mathbb R^{n+1}\), satisfies a local Lipschitz condition with respect to \(x\) and is continuous with respect to \(t\). A new method of constructing smooth global Lyapunov-Krasovskii type functions is given for the nonautonomous equation (*) without cycles. The five lemmas proved in the paper lead to the following theorem: If the equation (*) possesses a finite number of closed invariant sets, among them being neither cycles nor improper saddles and in neighborhoods of each invariant set the Krasovskij property occurs, then it is possible to construct a global Lyapunov-Krasovskij function of the class \(C^{\infty}\).
Reviewer: P.Talpalaru


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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