On the asymptotic stability of systems with impulses by the direct method of Lyapunov. (English) Zbl 0681.34042

Kulev, G. K.; Bainov, D. D.

J. Math. Anal. Appl. 140, No. 2, 324-340 (1989).

The paper deals with motions (x(t),y(t)) which satisfy a system of differential equations of the type (1) \(\dot x=f(t,x)+g(t,y),\) \(\dot y=h(t,x,y)\) for all \(t\neq t_ i(x,y)\) \((i=1,2,...)\), but perform certain jumps \((2)\quad \Delta x|_{t=t_ i(x,y)}=A_ i(x)+B_ i(y),\) \(\Delta y|_{t=t_ i(x,y)}=C_ i(x,y)\) when passing the hypersurfaces \(t=t_ i(x,y)\). In six theorems sufficient conditions for (local or global) asymptotic stability are given making use of the concepts of Lyapunov’s direct method for the trajectories of (1). A crucial assumption is that the distance of a motion from the origin (and also the value of its Lyapunov function) is non-increasing at each jump (2).

Reviewer: Wolfdietrich Müller

MSC:

34D20	Stability of solutions to ordinary differential equations
34D05	Asymptotic properties of solutions to ordinary differential equations

Keywords:

jumps; asymptotic stability; Lyapunov’s direct method; Lyapunov function

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