Controllability of linear impulsive stochastic systems in Hilbert spaces

This paper is concerned with the controllability of linear impulsive stochastic systems (LISSs) in Hilbert spaces. For this class of systems, the concepts of null controllability and approximate null controllability are introduced. To overcome the difficulties related, we construct the adjoint systems and the quasi-backward stochastic systems of LISSs. Necessary and sufficient conditions for the null controllability and the approximate null controllability are developed by utilizing our introduced quasi-backward stochastic systems. Furthermore, an equivalence is established between the null controllability of LISSs and some initial state of quasi-backward stochastic systems.