Stability of nonlinear impulsive differential equations with non-fixed moments of jumps

Abstract

This paper studies stability properties of the trivial solution to a system of nonlinear differential equations that undergo impulsive perturbations at non-fixed moments of time. We are motivated by modeling of networked control systems in which the communication between subsystems can be statedependent. This leads to the impulsive system with multiple impulsive time sequences and a distinct jump map for each sequence. Lyapunov-like theorems equipped with novel dwelltime conditions for global asymptotic stability of the origin have been proven. We treat the cases of a stable continuous dynamics that is being destabilized by impulsive perturbations, and vice versa, the case of unstable continuous dynamics that is being stabilized by impulses. Our results are less conservative comparing to the existing ones since we propose the concept of a candidate Lyapunov function with multiple nonlinear rate functions to characterize its behaviour during flows and jumps and account the influence of impulses for each impulsive time sequence separately. Also, we demonstrate the application of the results to stability analysis of impulsive differential equations with fixed moments of jumps and compare them with the existing ones.