Razumikhin-type theorems for hybrid system with memory

In this paper, we address the stability problem of hybrid systems with memory, which are newly developed to model hybrid dynamic systems affected by delays. For such systems, stability criteria are developed based on Razumikhin-type conditions, which are usually applied in the delay systems. We prove that the hybrid system with memory is uniformly globally pre-asymptotically stable by small gain theorem. Besides, a relaxed Razumikhin-type stability criterion is also proposed which does not require strict decrease of the Lyapunov–Razumikhin function during jumps. Several examples are given to illustrate the effectiveness and generality of the proposed results.