Contróle Exact De Léquation De La Chaleur

Lebeau, Gilles; Robbiano, Luc

doi:10.1080/03605309508821097

Cited by 589 publications

(773 citation statements)

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“…Note that in the proposition, we purposely do not impose any boundary condition at x n = 0 + on the function v here. The result of Lemma 2 in [LR95] is the counterpart of this proposition in the elliptic case.…”

Section: Proposition 324 We Havementioning

confidence: 69%

Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces

Rousseau

Robbiano

2010

Invent. math.

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Abstract. In (0, T ) × Ω, Ω open subset of R n , n ≥ 2, we consider a parabolic operator P = ∂ t − ∇ x δ(t, x)∇ x , where the (scalar) coefficient δ(t, x) is piecewise smooth in space yet discontinuous across a smooth interface S . We prove a global in time, local in space Carleman estimate for P in the neighborhood of any point of the interface. The "observation" region can be chosen independently of the sign of the jump of the coefficient δ at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface. In the high-frequency regime we use Calderón projectors. In the low-frequency regime we follow a more classical approach. Because of the parabolic nature of the problem we need to introduce Weyl-Hörmander anisotropic metrics, symbol classes and pseudo-differential operators. Each frequency regime and the associated technique require a different calculus. A global in time and space Carleman estimate on (0, T ) × M, M a manifold, is also derived from the local result.

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Section: Proposition 324 We Havementioning

confidence: 69%

Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces

Rousseau

Robbiano

2010

Invent. math.

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“…We recall that the internal null controllability of the linear heat equation, when the control acts on a subset of the domain, was established by Lebeau and Robbiano [10] and was later extended to some semilinear equation by Fursikov and Imanuvilov [4], in the sublinear case and by Barbu [2] and Fernandez-Cara [3], in the superlinear case.…”

Section: Dp(a T X) + µ(A)p(a T X) − K P(a T X) = F (Ax)+ M(x)umentioning

confidence: 99%

Local exact controllability of the age‐dependent population dynamics with diffusion

Aïnseba

Aniţa

2001

Abstract and Applied Analysis

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“…The first motivation for writing the proof came from the preprint version of [MZ04]. The final version of [MZ04] mentions that "a careful analysis of the method of proof in [LR95,LZ98] shows that it works if α > 1/2" without upper bound.…”

Section: Application To the Fractional Diffusionmentioning

confidence: 99%

“…In a first step, from the stationary condition in def.3, we deduce the observability of low-frequency modes over any positive time in the corresponding dynamics (this is the abstract version of sect.4 in [Mil05]). In a second step, using an abstract version of the iterative control strategy introduced by Lebeau and Robbiano in [LR95] (cf. sect.5 in [Mil05]), we prove the full null-controllability in arbitrarily small time.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian

Miller

2006

Math. Control Signals Syst.

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Abstract. This paper introduces a "spectral observability condition" for a negative self-adjoint operator which is the key to proving the null-controllability of the semigroup that it generates and to estimating the controllability cost over short times. It applies to the interior controllability of diffusions generated by powers greater than 1/2 of the Dirichlet Laplacian on manifolds, generalizing the heat flow. The critical fractional order 1/2 is optimal for a similar boundary controllability problem in dimension one. This is deduced from a subsidiary result of this paper, which draws consequences on the lack of controllability of some one dimensional output systems from Müntz-Szász theorem on the closed span of sets of power functions.In this paper, an observability condition on the spectral subspaces of a negative self-adjoint operator is introduced which ensures fast controllability, i.e. the semigroup generated by this operator is null-controllable in arbitrarily small time. In this asymptotic, it also ensures an upper bound for the controllability cost, i.e. the supremum, over every initial state with norm one, of the norm of the optimal input function which steers it to zero (cf. section 1). This spectral observability condition is the abstract version of a property proved for the Dirichlet Laplacian ∆ on a compact manifold observed on any non empty region in [LZ98, JL99] (cf.[Mil05] for non-compact manifolds).It applies to the semigroup generated by the fractional Laplacian on manifolds −(−∆) α as long as α > 1/2. This semigroup is widely used to describe physical systems exhibiting anomalous diffusions (cf. references in sect.2.1). Thus new interior null-controllability results for such fractional diffusions with non-constant coefficients in any dimension are deduced (the one dimension problem with constant coefficients and one dimensional input was recently considered in [MZ04]). In particular, as the control time T tends to 0, the controllability cost grows at most like C β exp(c β /T β ) where C β and c β are positive constants and β > 1/(2α − 1) (n.b. a lower bound of the same form with equality β = 1/(2α − 1) holds in the case α = 1 which corresponds to the heat equation). It is proved in the appendix that a similar one dimensional problem is not controllable from the boundary forThis last result is deduced from a more general remark of independent interest on the lack of controllability of any finite linear combination of eigenfunctions of systems with one dimensional input, based on the generalized Müntz theorem on the completeness of sets of exponentials.1. The main result in the abstract setting.After recalling the duality between controllability and observability for parabolic semigroups, this section states the main definition and theorem.1.1. The abstract setting. Let the generator A be a positive self-adjoint operator with domain D(A) on the Hilbert space H of states which we identify with its dual.2000 Mathematics Subject Classification. 93B05, 34K35, 58J35.

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Contróle Exact De Léquation De La Chaleur

Cited by 589 publications

References 3 publications

Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces

Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces

Local exact controllability of the age‐dependent population dynamics with diffusion

On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian

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