Impulse approximate controllability for stochastic evolution equations and its applications

This paper is concerned with impulse approximate controllability for stochastic evolution equations with impulse controls. As direct applications, we formulate captivating minimal norm and time optimal control problems; The minimal norm problem seeks to identify an optimal impulse control characterized by the minimum norm among all feasible controls, guiding the system’s solutions from an initial state within a fixed time interval toward a predetermined target while the minimal time problem is to find an optimal impulse control (among certain control constraint set), which steers the solution of the stochastic equation from a given initial state to a given target set as soon as possible. These problems, to the best of our knowledge, are among the first to discuss in the stochastic case.