Minimal time impulse control of the heat equation

TL;DR

This paper investigates the minimal time impulse control problem for the heat equation, establishing existence, uniqueness, and continuity of solutions, and analyzing the convergence of optimal controls under parameter variations.

Contribution

It introduces a novel framework for minimal time impulse control of the heat equation, including existence, uniqueness, and stability results for the optimal solutions.

Findings

Existence and uniqueness of optimal impulse control solutions.

Continuity of the minimal time function with respect to control bounds and impulse timing.

Convergence analysis of optimal controls as parameters vary.

Abstract

The paper is concerned with a kind of minimal time control problem for the heat equation with impulse controls. The purpose of such a problem is to find an optimal impulse control (among certain control constraint set) steering the solution of the heat equation from a given initial state to a given target set as soon as possible. We will first study the existence and uniqueness of optimal solution for this problem. In the formulation of this problem, there are two parameters: one is the upper bound of the control constraint and the other one is the moment of impulse time. Then, we will establish the continuity of the minimal time function of this problem with respect to the above mentioned two parameters. Moreover, the convergence of the optimal control is also discussed.