Novel Fixed-Time Stability Criteria for Discontinuous Nonautonomous Systems: Lyapunov Method With Indefinite Derivative.

This article considers a general class of nonautonomous discontinuous ordinary differential equations (ODE). By constructing the Filippov multimap, the fixed-time stability (FTS) problem of discontinuous ODE is transformed into that of differential inclusion (DI). In order to establish the FTS criteria of the zero solution for DI, the generalized Lyapunov function (LF) method is developed. The generalized LF of this article is relaxed to have an indefinite derivative for almost everywhere along the state trajectories of the system. However, the traditional LF is required to possess negative definite or semi-negative definite derivative for everywhere. As a result, several novel sufficient conditions for FTS are given. Moreover, the settling time of FTS is provided. Then, the theoretical results are applied to solve the fixed-time stabilization control problems of ball motion model and neural networks (NNs) with discontinuities. The developed LF method of FTS is extremely significant in the field of control engineering.