New Results for Hybrid Delay Systems With Initial Data: Razumikhin-Type Stability Criteria

This brief work focuses on the stability of a novel hybrid dynamical system formulated by hybrid phenomena, finite memory and initial data. To solve this issue, the main idea is to explore the pre-asymptotic stability and uniform pre-asymptotic stability of a hybrid delay model with initial data by developing the unique properties of hybrid memory domain, hybrid memory arc, generalized hybrid memory solution and continuation. Then, by introducing the Razumikhin-type function method, new pre-asymptotic stability and uniform pre-asymptotic stability theorems for hybrid delay systems with initial data are proposed in the framework of generalized hybrid solutions containing finite memory and initial data. Finally, an example analysis is conducted to reveal the validity of the proposed criteria.