On the stability of systems with impulsive effect (Q2880000)

This paper is devoted to the development of the direct Lyapunov method for non-autonomous systems of nonlinear differential equations with impulses at fixed time instants. The authors assume that the time intervals between sequential impulses are bounded by positive constants from above and below. In order to study the asymptotic behavior of solutions, a class of Lyapunov functions with discontinuities of the first kind at the time instants of the impulses is introduced. Sufficient conditions for the uniform asymptotic stability of the trivial solution are obtained in terms of an auxiliary function with negative time integral values along the solutions of the linearized system (Theorem 1). A result on instability is derived by means of a Lyapunov function that takes positive values in any neighborhood of the origin (Theorem 2). As an example, a system of differential equations with impulses is considered in the critical case of a pair of purely imaginary eigenvalues. A Lyapunov function is constructed by using the integral of the linearized system for this example.