Exhaustion approximation for the control problem of the heat or Schr\"odinger semigroup on unbounded domains

TL;DR

This paper demonstrates that controllability and control cost estimates for heat and Schrödinger equations on unbounded domains can be derived from the corresponding properties on bounded domains through exhaustion techniques.

Contribution

It establishes a method to transfer controllability results from bounded to unbounded domains using exhaustion sequences and control convergence.

Findings

Control solutions on unbounded domains can be approximated by solutions on bounded domains.

Control cost estimates are preserved in the limit from bounded to unbounded domains.

Controllability on unbounded domains can be inferred from bounded domain analysis.

Abstract

We consider the control problem of the heat equation on bounded and unbounded domains, and more generally the corresponding inhomogeneous equation for the Schr\"odinger semigroup. We show that if the sequence of null-controls associated to an exhaustion of an unbounded domain converges, then the solutions do in the same way, and that the control cost estimate carries over to the limiting problem on the unbounded domain. This allows to infer the controllability on unbounded domains by studying the control problem on a sequence of bounded domains.