Stability of Nonlinear Switched Fractional Differential Equations with Short Memory

Nonlinear switched systems, which combine multiple subsystems with a switching rule, have garnered significant research interest due to their complex stability properties. In this paper we consider the case where the switching times, the switching rule and the family of functions defining the subsystems are given initially. Note that the switching rule could be such that it is not activated at any initially given switching time. When the switching rule is activated, then a subsystem from the given family is chosen. We study the case where the nonlinear subsystems consist of fractional differential equations. To be more concise we apply generalized Caputo fractional derivatives with respect to other functions. Lyapunov functions are used to analyze stability, and several sufficient conditions are obtained. The influence of the switching rule on the stability property of the solutions is discussed and illustrated with examples. It is shown that, in spite of the solutions of some of the subsystems being unstable, the zero solution of the switched system could be stable.