Equivalence between minimal time and minimal norm control problems for the heat equation

TL;DR

This paper explores the relationship between minimal time and minimal norm control problems for heat equations with various target sets, revealing conditions under which these problems are equivalent or differ.

Contribution

It extends previous work by analyzing arbitrary convex target sets, showing how the properties of minimal norm and time functions vary with target location.

Findings

Minimal norm and time functions are inverses when target is origin or centered ball.

For non-origin targets, minimal norm may be non-monotonous.

Range of minimal time function may be disconnected for arbitrary targets.

Abstract

This paper presents the equivalence between minimal time and minimal norm control problems for internally controlled heat equations. The target is an arbitrarily fixed bounded, closed and convex set with a nonempty interior in the state space. This study differs from [G. Wang and E. Zuazua, \textit{On the equivalence of minimal time and minimal norm controls for internally controlled heat equations}, SIAM J. Control Optim., 50 (2012), pp. 2938-2958] where the target set is the origin in the state space. When the target set is the origin or a ball, centered at the origin, the minimal norm and the minimal time functions are continuous and strictly decreasing, and they are inverses of each other. However, when the target is located in other place of the state space, the minimal norm function may be no longer monotonous and the range of the minimal time function may not be connected. These cause the main difficulty in our study. We overcome this difficulty by borrowing some idea from the classical raising sun lemma (see, for instance, Lemma 3.5 and Figure 5 on Pages 121-122 in [E. M. Stein and R. Shakarchi, \textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}, Princeton University Press, 2005]).