Null Controllability in Unbounded Domains for the Semilinear Heat Equation with Nonlinearities Involving Gradient Terms

Cabanillas, V. R.; Menezes, S. B. de; Zuazua, Enrique

doi:10.1023/a:1017515027783

Cited by 45 publications

(38 citation statements)

References 9 publications

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“…This refines the result of [3] in the linear case. (However note that the total measure of the control region ω is still infinite in this case.…”

Section: Examplesupporting

confidence: 86%

“…However Cabanillas, De Menezes and Zuazua [3] proved that (2.4) is true with ρ ≡ 1 when Ω \ ω is bounded. Hence our estimates are intermediate between the negative result when ω is bounded and the positive result when Ω \ ω is bounded.…”

Section: Null Controllability Results In One Space Dimensionmentioning

confidence: 99%

“…Very recently, Cabanillas, De Menezes and Zuazua [3] studied the problem of null controllability when the domain Ω is unbounded but when the control acts on a large control region ω: under the assumption Ω \ ω is bounded, they obtained positive results: given T > 0 and u 0 ∈ L 2 (Ω), there exists a control f ∈ L 2 (0, T ; L 2 (Ω)) such that the solution of (1.1) reaches the rest at time T . (More generally, they obtained similar results for the semilinear heat equation, with nonlinearities involving gradient terms.…”

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confidence: 98%

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

2004

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Abstract. Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes andZuazua, we study the null controllability of the heat equation in unbounded domains, typically R+ or R N . Considering an unbounded and disconnected control region of the form ω := ∪nωn, we prove two null controllability results: under some technical assumption on the control parts ωn, we prove that every initial datum in some weighted L 2 space can be controlled to zero by usual control functions, and every initial datum in L 2 (Ω) can be controlled to zero using control functions in a weighted L 2 space. At last we give several examples in which the control region has a finite measure and our null controllability results apply.Mathematics Subject Classification. 35K05, 93B05, 93B07.

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“…This refines the result of [3] in the linear case. (However note that the total measure of the control region ω is still infinite in this case.…”

Section: Examplesupporting

confidence: 86%

Section: Null Controllability Results In One Space Dimensionmentioning

confidence: 99%

mentioning

confidence: 98%

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

2004

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“…In [MZ03], it is mentioned that their approach does not apply to unbounded Euclidean domains (indeed Weyl's asymptotics for eigenvalues and Russell's construction of biorthogonal functions are used in [LR95]) and that the only result available on these specific properties is that they hold when M is a domain of the Euclidean space with the flat metric and the exterior of Ω is bounded (cf. [CDMZ01]). To this open problem, this article contributes the adaptation of the Lebeau-Robbiano approach to a non-compact M , the extension of the sufficient condition in [CDMZ01] and the necessary condition in [Mil04b] to a non-Euclidean M , and a finer sufficient condition for a homogeneous M which applies even if the exterior of Ω is not compact.…”

mentioning

confidence: 99%

“…[CDMZ01]). To this open problem, this article contributes the adaptation of the Lebeau-Robbiano approach to a non-compact M , the extension of the sufficient condition in [CDMZ01] and the necessary condition in [Mil04b] to a non-Euclidean M , and a finer sufficient condition for a homogeneous M which applies even if the exterior of Ω is not compact.…”

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confidence: 99%

Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds

Miller¹

2005

Mathematical Research Letters

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Abstract. This article concerns some quantitative versions of unique continuation known as observability inequalities. One of them is a lower bound on the spectral projectors of the Dirichlet Laplacian which generalizes the unique continuation of an eigenfunction from any open set Ω. Another one is equivalent to the interior null-controllability in time T of the heat equation with Dirichlet condition (the input function is a source in (0, T ) × Ω). On a compact Riemannian manifolds, these inequalities are known to hold for arbitrary T and Ω.This article states and links these observability inequalities on a complete non-compact Riemannian manifold, and tackles the quite open problem of finding which Ω and T ensure their validity. It proves that it is sufficient for Ω to be the exterior of a compact set (for arbitrary T ), but also illustrates that this is not necessary. It provides a necessary condition saying that there is no sequence of balls going infinitely far "away" from Ω without "shrinking" in a generalized sense (depending on T ) which also applies when the distance to Ω is bounded.

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On the Optimality of the Observability Inequalities for Kirchhoff Plate Systems with Potentials in Unbounded Domains

Zhang

Zuazua

2008

Hyperbolic Problems: Theory, Numerics, Applications

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Null Controllability in Unbounded Domains for the Semilinear Heat Equation with Nonlinearities Involving Gradient Terms

Cited by 45 publications

References 9 publications

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

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

Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds

On the Optimality of the Observability Inequalities for Kirchhoff Plate Systems with Potentials in Unbounded Domains

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