Necessary and sufficient stability conditions for invariant sets of nonlinear impulsive systems

Abstract

Ignat'ev UDC 539.36Necessary and sufficient conditions for the uniform asymptotic stability of the invariant set of a nonlinear impulsive system are established Introduction. Systems of impulsive differential equations adequately model many real physical systems subjected to short-term forces at certain instants of time. Impulsive systems are of interest due primarily to their numerous applications, including those in mechanics. For example, such systems can be used in robotics, in space engineering, in modeling vibropercussion machines, buffer units, etc. Intensive development of the theory of impulsive systems and their successful application in applied problems require efficient stability criteria for solutions of such systems. The studies [1, 3, 4, 6, 7, 9-17, etc.] contributed a lot to the development of the theory of stability of impulsive systems. The Lyapunov-function method is one of the most efficient techniques for stability analysis of nonlinear impulsive systems.The overwhelming majority of published studies that use the direct Lyapunov method analyze the sufficient stability conditions for solutions of impulsive systems, giving little attention to the issue of existence of Lyapunov functions. Whether Lyapunov functions exist is the key question in the direct Lyapunov method because answering it would indicate whether it is expedient to apply the Lyapunov-function method. The necessary conditions for the stability of solutions of impulsive systems with respect to all variables were established in [2,11].The purpose of this paper is to derive the necessary and sufficient conditions for the uniform asymptotic stability of the invariant set of a nonlinear impulsive system. Problem Statement. Consider a system of differential equations with impulsive effect at fixed times:

Necessary conditions for the asymptotic stability of systems of differential equations with pulse action on surfaces

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Existence Results for Systems of Nonlinear Second-Order and Impulsive Differential Equations with Periodic Boundary

“…Certain of the relevant results are generalized in monograph [4] using piecewise differentiable auxiliary functions. The works [5,6] show the universality of the direct Lyapunov method in this class of auxiliary functions. In [7], stability conditions for solutions of a nonlinear system with impulse action were obtained based on two auxiliary functions.…”