Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations

Fattorini, H. O.; Russell, David L.

doi:10.1090/qam/510972

Cited by 153 publications

(176 citation statements)

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“…Fattorini and D.L. Russell (see [19,20]) through the moment method. The controllability of the N -dimensional case, still for the scalar equation (n = 1), has been established later by G. Lebeau and L. Robbiano in [31] and by A. Fursikov and O. Yu.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Unlike the scalar case, from the results in [10] we infer that the approximate and null boundary controllability properties for non-scalar parabolic systems are, in general, not equivalent. In [21], [6] and [10], the authors use the moment method (see [19,20]) to prove the positive null controllability result at time T . They carry out a study on bounds of biorthogonal families to exponentials associated to complex sequences.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence

Khodja

Benabdallah

González-Burgos

et al. 2016

Journal of Mathematical Analysis and Applications

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We consider the null controllability problem for two coupled parabolic equations with a space-depending coupling term. We analyze both boundary and distributed null controllability. In each case, we exhibit a minimal time of control, that is to say, a time T0 ∈ [0, ∞] such that the corresponding system is null controllable at any time T > T0 and is not if T < T0. In the distributed case, this minimal time depends on the relative position of the control interval and the support of the coupling term. We also prove that, for a fixed control interval and a time τ0 ∈ [0, ∞], there exist coupling terms such that the associated minimal time is τ0.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence

Khodja

Benabdallah

González-Burgos

et al. 2016

Journal of Mathematical Analysis and Applications

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“…This paper is devoted to characterizing the class of initial data u 0 such that the solution of (1.1) may be driven to zero in time T by means of an L 2 control v. In the case of bounded domains, using Fourier series expansion, the control problem may be reduced to a moment problem (see for instance [5]). However, since we are working on R + , we cannot use Fourier series.…”

Section: Introduction Problem Formulationmentioning

confidence: 99%

On the lack of null-controllability of the heat equation on the half-line

Micu

Zuazua

2000

Trans. Amer. Math. Soc.

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Abstract. We consider the linear heat equation on the half-line with a Dirichlet boundary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the L 2 boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials {e jt } j≥1 in which the usual summability condition on the inverses of the eigenvalues does not hold. Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.

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“…By (b) this implies Neumann null-controllability for Q 0 (alternatively, this is already given for the «-ball by results in [1]) and so, by (c), for £1 itself (alternatively, this is given by results in [4]). Then (b) implies well-posedness of the restricted observation/prediction problem for the heat equation in ÇI and so, by the reduction (a), of the general problem.…”

mentioning

confidence: 91%