Advantages for controls imposed in a proper subset

TL;DR

This paper investigates time optimal control problems for heat equations, demonstrating that controls on proper subsets of the domain exhibit existence and bang-bang properties, unlike controls on the entire domain, and reveals a new eigenfunction property.

Contribution

It establishes the conditions under which existence and bang-bang properties hold for controls on subsets, and introduces a novel eigenfunction property for the Laplacian.

Findings

Existence of time optimal controls on proper subsets

Bang-bang property holds on proper subsets

A new eigenfunction property for the Laplacian

Abstract

In this paper, we study time optimal control problems for heat equations on $\Omega\times \mathbb{R}^+$. Two properties under consideration are the existence and the bang-bang properties of time optimal controls. It is proved that those two properties hold when controls are imposed on some proper subsets of $\Omega$; while they do not stand when controls are active on the whole $\Omega$. Besides, a new property for eigenfunctions of $-\Delta$ with Dirichlet boundary condition is revealed.