Null controllability of the heat equation in unbounded domains by a finite measure control region

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 exact 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.

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Null controllability of a nonlinear heat equation

Abstract and Applied Analysis ◽

10.1155/s1085337502002865 ◽

2002 ◽

Vol 7 (7) ◽

pp. 375-383 ◽

Cited By ~ 2

Author(s):

G. Aniculăesei ◽

S. Aniţa

Keyword(s):

Heat Equation ◽

Dirichlet Boundary ◽

Null Controllability ◽

Carleman Estimates ◽

Nonlinear Heat Equation ◽

Homogeneous Dirichlet Boundary Condition ◽

Linearized System ◽

Kakutani Fixed Point ◽

Exact Null Controllability ◽

Kakutani Fixed Point Theorem

We study the internal exact null controllability of a nonlinear heat equation with homogeneous Dirichlet boundary condition. The method used combines the Kakutani fixed-point theorem and the Carleman estimates for the backward adjoint linearized system. The result extends to the case of boundary control.

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Null controllability of the heat equation in pseudo-cylinders by an internal control

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020048 ◽

2020 ◽

Vol 26 ◽

pp. 122

Author(s):

Jon Asier Bárcena-Petisco

Keyword(s):

Heat Equation ◽

Internal Control ◽

Cylindrical Part ◽

Null Controllability ◽

Carleman Estimate ◽

Main Difficulty ◽

Absorption Properties ◽

Weighted Function ◽

Cylindrical Structure ◽

Linear Heat

In this paper we prove the null controllability of the heat equation in domains with a cylindrical part and limited by a Lipschitz graph. The proof consists mainly on getting a Carleman estimate which presents the usual absorption properties. The main difficulty we face is the loss of existence of the usual weighted function in C2 smooth domains. In order to deal with this, we use its cylindrical structure and approximate the system by the same system stated in regular domains. Finally, we show some applications like the controllability of the semi-linear heat equation in those domains.

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