A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

TL;DR

This paper establishes a uniform upper bound on control costs for semilinear heat equations across increasing domains, enabling the proof of global null controllability in unbounded space.

Contribution

It introduces a novel uniform bound on control costs for semilinear heat equations on expanding domains, facilitating controllability in unbounded Euclidean space.

Findings

Uniform upper bound on control costs for increasing domains

Achieved null controllability in R^N for semilinear heat equations

Overcame compactness issues in unbounded domain control problems

Abstract

In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space R^N . As an application, we then show the exactly null controllability for this semilinear heat equation in R^N . The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null controllability for the semilinear heat equation in R^N . This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null controllability for nonlinear PDEs in generally unbounded domains.