Input-to-state stability of impulsive stochastic infinite dimensional systems with Poisson jumps

Abstract

This paper is concerned with the input-to-state stability (ISS) of mild solutions to a class of impulsive stochastic infinite dimensional systems with Poisson jumps (ISISP). By using a Yosida strong solutions approximation approach for mild solutions of ISISP and the Itô’s formula, several Lyapunov-based sufficient conditions ensuring the ISS of ISISP with stabilizing and destabilizing impulses are established, respectively. The main merits of the sufficient conditions are that (i) the ISS-Lyapunov function can be non-exponential; (ii) average dwell-time condition is imposed to restrict the occurrence frequency of impulses; (iii) the effects of destabilizing or stabilizing impulses can be explicitly shown compared with the existing works about the stability of mild solutions to impulsive stochastic infinite dimensional systems by the fixed point approach. Moreover, when the ISS-Lyapunov function is exponential, we present a more general result which is suitable for destabilizing and stabilizing impulses simultaneously. Finally, examples of stochastic heat equations are provided to illustrate the proposed results.