Stability for Switched Fractional Differential Equations with Random Switching Times

Stability analysis of switched nonlinear fractional differential equations is considered in this paper. The main characteristic of our problem is random switching times when the switching rule is activated. Here, the switching rule is independent of the state of the system, but it depends on the random activation points. The dynamics between two consecutive switching times are described by nonlinear fractional differential equations with the Caputo fractional derivative with respect to another function. The lower limit of the fractional derivative changes at each switching time, and the waiting time between consecutive switching times is an Erlang-distributed random variable. We define p-moment stability, and we obtain some sufficient conditions; the study is based on an application of a Lyapunov function method. We consider both the case of Lyapunov functions with a negative fractional derivative and the case with a positive fractional derivative. It is of interest to note that the applied function in the Caputo-type fractional derivative has a huge influence on the stability behavior of the system.