The Input-to-State Stability in Probability for Constrained Delay Systems With Stochastic Delayed Impulses

This paper investigates the problem of the input-to-state stability in probability (ISSiP) for a family of constrained delay systems with stochastic delayed impulses. A new type of admissible edge-dependent average impulsive interval (AED-AII) impulsive signal is presented, in which the concepts of AED-AII impulsive intensity and AED-AII impulsive density are firstly presented to comprehensively describe the impulsive signals. Under the constraint of stochastic impulses, we study the ISSiP of impulsive systems in two different scenarios. One scenario is that the impulsive intensity is stochastic and the impulsive density is limited by a deterministic impulsive interval. Another scenario is that both impulsive intensity and density are subject to stochastic constraints, and the impulsive interval is satisfied by a renewal process. Some novel AED-AII-based criteria for ISSiP are deduced, respectively, where a relationship between impulsive density, impulsive intensity, the jumping probability of impulse mode and Lyapunov function decay rate is established. Then, we extend this study to constrained impulsive switched systems and obtain a class of AED-based criteria to guarantee the ISSiP property under different impulsive conditions. Finally, examples are given to depict the validity of the proposed stability criteria. Note to Practitioners—The problem discussed in this paper is the stability problem of a set of constrained delay systems. Under the constraint of stochastic impulses, for the first time, the intensity and density of impulsive signals are characterized from the perspective of edge-dependent, and input-to-state stability in probability is studied, which is then extended to impulsive switched systems. We propose AED-AII to investigate the different situations of impulsive intensity and impulsive density under random constraints, in order to ensure the ISSiP of the system. Considering the structure and performance of circuit systems and robot manipulators, such as non-autonomous Chua’s circuits with nonlinear resistors, external inputs, and random noise, this method can be applied to systems with random noise.