The stability of a solution of a system of differential equations with discontinuous right-hand sides

Abstract

The stability of the periodic solution of the system of equations dz dt = f (z,t) with discontinous periodic [ f( z, t + r) = f( z, t)] right-hand sides [ z and f are n-dimensional vector columns with coordinates z i and f i ( i = 1, ·, n)] has been investigated by Aizerman and Gantmakher [1]. Establishing what should be understood by the linear approximation in this “discontinuous” case, the authors have proved theorems analogous to those of Liapunov. The present paper deals with the stability of any solution (periodic or nonperiodic) of system (0.1) with discontinuous nonperiodic right-hand sides. For this, use is made of the condition for the discontinuities of the solution of the linear approximation introduced in paper [1] for periodic systems. Two criteria of stability are established which are generalizations of the corresponding theorems of Persidskii [2] and Perron [3], proved by these authors for continuous systems.