Volume 28, 2022
|Number of page(s)||31|
|Published online||21 January 2022|
A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application
School of Mathematics and Statistics, Wuhan University,
2 Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China.
* Corresponding author: email@example.com
Accepted: 4 January 2022
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.
Mathematics Subject Classification: 35K05 / 93B07 / 93C20
Key words: Semilinear heat equation / null-controllability / uniform cost / equidistributed set
© The authors. Published by EDP Sciences, SMAI 2022
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