A block moment method to handle spectral condensation phenomenon in parabolic control problems

Benabdallah, Assia; Boyer, Franck; Morancey, Morgan

doi:10.5802/ahl.45

Cited by 27 publications

(51 citation statements)

References 41 publications

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“…we recover in (1.21) the same behavior when λ → +∞ as the one given in [4,10,23]. In more general situations, the estimate given in Theorem 1.3 is sub-optimal.…”

Section: Existence and Estimates Of A Biorthogonal Family To The Exponentialssupporting

confidence: 81%

“…With such weak gap condition, uniform estimates (with respect to different parameters) are given in [10] as a byproduct of the block moments method. We also mention the recent paper [23] where precise estimates, including time-dependency of the constants, are given in this setting.…”

Section: Existence and Estimates Of A Biorthogonal Family To The Exponentialsmentioning

confidence: 99%

“…In more general situations, the estimate given in Theorem 1.3 is sub-optimal. However, with the spectral analysis of Theorem 1.2, let us notice that the estimates given in [4,10,23] would not allow to prove Theorem 1.1. In fact, we will deal with the sequence of families Λ n which satisfies, for each n, a weak gap condition but with a parameter p going to +∞ with n. Thus, it is crucial to notice that Theorem 1.3 do not require any gap-type condition and that the constants appearing in estimate (1.21) only depend on the asymptotic of the counting function.…”

Section: Existence and Estimates Of A Biorthogonal Family To The Exponentialsmentioning

confidence: 99%

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Analysis of the null controllability of degenerate parabolic systems of Grushin type via the moments method

2021

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In this article we compute the exact value of the minimal null control time for the Grushin equation controlled on a strip. This result is already known from a recent work by K. Beauchard, J. Dardé and S. Ervedoza but we propose a different approach based on the moments method. Our approach requires a careful spectral analysis of a truncated harmonic oscillator. We have also extended known results on biorthogonal families to real exponentials in the absence of a gap condition in order to obtain uniform estimates with respect to the asymptotic beahviour of the associated counting function. As a byproduct of our approach we also characterize the minimal null control time for systems of coupled Grushin equations with a distributed control on a strip or with a boundary control.

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“…we recover in (1.21) the same behavior when λ → +∞ as the one given in [4,10,23]. In more general situations, the estimate given in Theorem 1.3 is sub-optimal.…”

Section: Existence and Estimates Of A Biorthogonal Family To The Exponentialssupporting

confidence: 81%

Section: Existence and Estimates Of A Biorthogonal Family To The Exponentialsmentioning

confidence: 99%

Section: Existence and Estimates Of A Biorthogonal Family To The Exponentialsmentioning

confidence: 99%

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Analysis of the null controllability of degenerate parabolic systems of Grushin type via the moments method

2021

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“…As a consequence, we will see that a minimal time of control T 0 ∈ [0, +∞], depends simultaneously of condensation of eigenvalues and associated eigenfunctions of (L * , D(L * )). To this end, we will use the block methods moment developed by A. Benabdallah, F. Boyer, M. Morancey (in [5]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Boundary null-controllability of two coupled parabolic equations: simultaneous condensation of eigenvalues and eigenfunctions

Samb

2021

ESAIM: COCV

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Let the matrix operator $L=D\partial_{xx}+q(x)A_0 $, with $D=diag(1,\nu)$, $\nu\neq 1$, $q\in L^{\infty}(0,\pi)$, and $A_0$ is a Jordan block of order $1$. We analyze the boundary null controllability for the system $y_{t}-Ly=0$. When $\sqrt{\nu} \notin \mathbb{Q}_{+}^*$ and $q$ is constant, $q=1$ for instance, there exists a family of root vectors of $(L^*,\mathcal{D}(L^*))$ forming a Riesz basis of $L^{2}(0,\pi;\mathbb{R}^2 )$. Moreover in \cite{JFA14} the authors show the existence of a minimal time of control depending on condensation of eigenvalues of $(L^*,\mathcal{D}(L^*))$, that is to say the existence of $T_0(\nu)$ such that the system is null controllable at time $T > T_0(\nu)$ and not null controllable at time $T < T_0(\nu)$. In the same paper, the authors prove that for all $\tau \in [0, +\infty]$, there exists $\nu \in ]0, +\infty[$ such that $T_0(\nu)=\tau$. When $q$ depends on $x$, the property of Riesz basis is no more guaranteed. This leads to a new phenomena: simultaneous condensation of eigenvalues and eigenfunctions. This condensation affects the time of null controllability.

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“…Since then, the boundary controllability results for various parabolic problems was investigated (see [4,5,6,8,7,18,26]).…”

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confidence: 99%

Boundary controllability for a coupled system of degenerate/singular parabolic equations

Allal

Hajjaj

Salhi

et al. 2022

EECT

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<p style='text-indent:20px;'>In this paper we study the boundary controllability for a system of two coupled degenerate/singular parabolic equations with a control acting on only one equation. We analyze both approximate and null boundary controllability properties. Besides, we provide an estimate on the null-control cost. The proofs are based on a detailed spectral analysis and the use of the moment method by Fattorini and Russell together with some results on biorthogonal families.</p>

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A block moment method to handle spectral condensation phenomenon in parabolic control problems

Cited by 27 publications

References 41 publications

Analysis of the null controllability of degenerate parabolic systems of Grushin type via the moments method

Analysis of the null controllability of degenerate parabolic systems of Grushin type via the moments method

Boundary null-controllability of two coupled parabolic equations: simultaneous condensation of eigenvalues and eigenfunctions

Boundary controllability for a coupled system of degenerate/singular parabolic equations

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