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Lyapunov functions and comparison theorems for control systems with impulsive actions. (Russian. English summary) Zbl 1333.34024

Summary: We extend the results of E. A. Panasenko and E. L. Tonkov [Proc. Steklov Inst. Math. 268, S204-S221; translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 15, No. 3, 185–201 (2009; Zbl 1302.93182)] to differential equations and control systems with impulsive actions. In terms of Lyapunov functions and the Clarke derivative we obtain comparison theorems for systems with impulsive effect. We consider the set \(\mathfrak M\doteq\bigl\{(t,x)\in[t_0,+\infty)\times\mathbb R^n\colon x\in M(t)\bigr\}\), defined by continuous function \(t \to M(t)\), where for every \(t\in \mathbb R\) the set \(M(t)\) is nonempty and compact. We obtain conditions for the positive invariance of this set, the uniform Lyapunov stability and the uniform asymptotic stability. We make a comparison with the researches of other authors who have considered the zero solution stability for similar systems.

MSC:

34A37 Ordinary differential equations with impulses
34A60 Ordinary differential inclusions
49J15 Existence theories for optimal control problems involving ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34H05 Control problems involving ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Citations:

Zbl 1302.93182
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