Existence and stability of impulsive stochastic coupled systems in infinite dimensions

Abstract

This paper considers the exponential stability of a class of infinite‐dimensional impulsive stochastic coupled systems. With the help of generalized Itô's formula for the mild solution of infinite‐dimensional systems, we avoid limiting the domain of the mild solution. Then we use the combination of the Lyapunov function and graph theory to construct the Lyapunov function of the systems; the criteria of ‐th moment exponential stability are obtained, which is related to the average impulsive interval and the connectivity of impulsive stochastic systems. In addition, noting that the existence may be affected by impulsive effects and stochastic perturbations, using the graph theory and the principle of contraction mapping, we get the condition that guarantees the existence and uniqueness, which is also related to the structure of the networks. Finally, we consider the stability of impulsive stochastic coupled heat equations and neural networks with reaction diffusion and give some numerical simulations to verify the theoretical results.