Minimal Time Impulse Control of an Evolution Equation

This paper is concerned with a kind of minimal time control problem for a linear evolution equation with impulse controls. Each problem depends on two parameters: the upper bound of the control constraint and the moment of impulse time. The purpose of such a problem is to find an optimal impulse control (among certain control constraint set), which steers the solution of the evolution equation from a given initial state to a given target set as soon as possible. In this paper, we study the existence of optimal control for this problem; by the geometric version of the Hahn–Banach theorem, we show the bang–bang property of optimal control, which leads to the uniqueness of the optimal control; we also establish the continuity of the minimal time function of this problem with respect to the above mentioned two parameters, and discuss the convergence of the optimal control when the two parameters converge.