Input-to-state stability for impulsive switched nonlinear systems with unstable subsystems

In this article, the input-to-state stability is investigated for impulsive switched systems. By means of the Lyapunov function and the average impulsive switched interval approach, the input-to-state stability properties are derived under the condition that all subsystems are stable, all subsystems are unstable and some subsystems are unstable. It is shown that if the continuous subsystems all have input-to-state stability and though the impulsive effects are destabilizing, the system has input-to-state stability with respect to a lower bound of the average impulsive switched interval. Moreover, if all the subsystems do not have input-to-state stability, the impulsive effects can still successfully stabilize the system but for an upper bound of the average impulsive switched interval. However, it is unveiled that if some continuous subsystems are not input-to-state stability, the impulsive effects can successfully stabilize the system for a lower bound of the average impulsive switched interval under specific conditions. It is worth noting that we introduce multiple jumps in this paper. Finally, three examples are illustrated with their simulations to manifest the validity of the main results.