Minimal time control for the heat equation with multiple impulses

This paper investigates a minimal time control problem for the heat equation with multiple impulse controls. We first establish the maximum principles for this problem and then prove the equivalence between the minimal time impulse control problem and its corresponding minimal norm impulse control problem. Extending our analysis to the minimal time function itself, we examine its analytical properties (particularly continuity and monotonicity) with respect to the control constraint. This leads to a notable discovery: despite being continuous and generally monotonic, the function may fail to be strictly decreasing.