Sufficient Conditions for Local and Global Stability of Time-Varying Sets

Abstract

Theorems giving the sufficient conditions for local and global stability of a set (in general it is assumed to be time-varying) for a wide class of non linear systems are stated. The systems for which these theorems can be applied include those described by non-autonomous differential equations, differential inequalities, and differential inclusions. Theorems are formulated for systems with existing nonnegative Lyapunov function V related to the set whose stability is examined. The sufficient conditions are formulated in terms of restrictions imposed on V' or ∂V∂x and right-hand side of system subjected to investigation. For the case of autonomous system of differential equations with bounded trajectories the result is a corollary of LaSalle invariance principle. Two examples where obtained theorems are applicable are discussed. The numerical simulation results for these examples are presented.