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

Lijuan Wang,Can Zhang

doi:10.1051/cocv/2022001

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 ℝN. As an application, we then show the exact null-controllability for this semilinear heat equation in ℝN. 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 ℝN. 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.

Highlights

Introduction and main resultsThis paper is concerned with the null control costs for semilinear heat equations on a sequence of increasing bounded domains in RN, when the controls act on the interior subsets of these domains
We show the exact null-controllability for this semilinear heat equation in RN
We have proved in [8] the observability inequality and null-controllability on such kind of sets for the linear heat equation with time and space dependent potentials in RN

Summary

Introduction and main results

This paper is concerned with the null control costs for semilinear heat equations on a sequence of increasing bounded domains in RN (with N ∈ N), when the controls act on the interior subsets of these domains. The first main result of this paper concerning the uniform upper bound on costs of controlling a semilinear heat equation on increasing large domains can be stated as follows. We refer the reader to Remark 1.7 of [8] for the difficulty Another motivation of this paper is to establish the null-controllability for the semilinear heat equation in the whole space RN , when the control is acted on an equidistributed set. The authors of [7] studied the approximate controllability of a semilinear heat equation in an unbounded domain O of RN , with control only acted in an open and nonempty subset, by an approximation method They first considered the approximate controllability problem in bounded domains of the form On O Bn, where Bn denotes the ball centered at the origin and of radius n; and they showed that the controls proposed in [11] restricted to On converge in some sense to a desired approximate control in O, as n goes to infinity.

The linear case

Quantitative estimates of unique continuation

A UNIFORM BOUND ON COSTS OF CONTROLLING SEMILINEAR HEAT EQUATIONS

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