Generalized Lipschitz Stability for Switched Differential Equations with Riemann–Liouville Fractional Derivatives with Respect to Another Function

Switched fractional differential equations with Riemann–Liouville fractional derivatives with respect to another function (RLDFs) with orders between zero and one are studied. We provide a detailed algorithm for constructing the solution of the studied switched fractional system, and we illustrate with several examples the influence on both the switching rules and the applied function in the fractional derivative on the behavior of the solution. We define generalized Lipschitz stability in time for the studied system, and this type of stability guarantees that the solutions will be close to the initial values on intervals excluding the points of singularities, i.e., the initial time point and the switching time points. We prove auxiliary results for Lyapunov functions and their RLDFs, including quadratic Lyapunov functions. These results are applied to study generalized Lipschitz stability in time.