Approximation of time optimal controls for heat equations with perturbations in the system potential

TL;DR

This paper investigates how small perturbations in the potential of a heat equation affect the optimal control and timing, establishing approximation results and constructing equivalent control problems with similar optimal times and controls.

Contribution

It introduces an approximation framework for time optimal controls of heat equations with perturbed potentials, extending existing equivalence theorems to this setting.

Findings

Optimal time and control are close under small potential perturbations.

Constructed alternative control problems share the same optimal time as the original.

Established the use of an equivalence theorem for minimal norm and minimal time controls.

Abstract

In this paper, we study a certain approximation property for a time optimal control problem of the heat equation with $L^\infty$-potential. We prove that the optimal time and the optimal control to the same time optimal control problem for the heat equation, where the potential has a small perturbation, are close to those for the original problem. We also verify that for the heat equation with a small perturbation in the potential, one can construct a new time optimal control problem, which has the same target as that of the original problem, but has a different control constraint bound from that of the original problem, such that the new and the original problems share the same optimal time, and meanwhile the optimal control of the new problem is close to that of the original one. The main idea to approach such approximation is an appropriate use of an equivalence theorem of minimal norm and minimal time control problems for the heat equations under consideration. This theorem was first established by G.Wang and E. Zuazua in [20] for the case where the controlled system is an internally controlled heat equation without the potential and the target is the origin of the state space.