No abstract

In this paper, we study a class of stochastic partial differential equations with Poisson jumps, which is more realistic for establishing mathematical models since it has been widely applied in many fields. Under a reasonable condition, we not only establish the existence and uniqueness of the mild solution for the investigated system but also prove that it is pth moment exponentially stable by using the fixed point theory. Then, based on the well-known Borel-Cantelli lemma, further, we prove that the mild solution is almost surely pth moment exponentially stable. Our results improve and generalize those given in the previous literature, in particular, the Lyapunov direct method and successive approximation method. Finally, we give an example to illustrate the effectiveness of the obtained results.

The mean‐square exponential stabilization and stochastically asymptotical stabilization for a class of stochastic functional differential systems is studied. Based on the definitions of derivatives of functionals, the generalized Itô operator for functionals and compound functions is established. Furthermore, the novel Lyapunov functionals and the induced steepest descent feedback controls are constructed to obtain such stability conditions that are weaker than the classical ones. Together with the generalized Itô operator and the steepest descent feedback controls, mean‐square exponential stabilization and stochastically asymptotical stabilization for stochastic functional differential systems are investigated, respectively. As applications, two examples are given to illustrate the derived results: one is a stochastic mass–spring–damper model and the other is a numerical example with simulation figures.