Indirect stabilization of locally coupled wave-type systems

Alabau‐Boussouira, Fatiha; Léautaud, Matthieu

doi:10.1051/cocv/2011106

Cited by 57 publications

(38 citation statements)

References 27 publications

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“…To our knowledge, the only results on this issue are the one of [ABL12] and [AB12]. Let us also mention [Oli13] for related questions for the approximate controllability problem.…”

mentioning

confidence: 99%

Sharp Estimates of the One-Dimensional Boundary Control Cost for Parabolic Systems and Application to the $N$-Dimensional Boundary Null Controllability in Cylindrical Domains

Benabdallah¹,

Boyer²,

González-Burgos³

et al. 2014

SIAM J. Control Optim.

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Abstract. In this paper we consider the boundary null-controllability of a system of n parabolic equations on domains of the form Ω = (0, π) × Ω 2 with Ω 2 a smooth domain of R N −1 , N > 1. When the control is exerted on {0} × ω 2 with ω 2 ⊂ Ω 2 , we obtain a necessary and sufficient condition that completely characterizes the nullcontrollability. This result is obtained through the Lebeau-Robbiano strategy and require an upper bound of the cost of the one-dimensional boundary null-control on (0, π). This latter is obtained using the moment method and it is shown to be bounded by Ce C/T when T goes to 0 + . Key words. Parabolic systems, Boundary Controllability, Biorthogonal families, Kalman Rank condition.AMS subject classifications. 93B05, 93C05, 35K05.1. Introduction. The controllability of systems of n partial differential equations by m < n controls is a relatively recent subject. We can quote [LZ98], [dT00], [BN02] among the first works. More recently in [AKBDGB09b], with fine tools of partial differential equations, the socalled Kalman rank condition, which characterizes the controllability of linear systems in finite dimension, has been generalized in view of the distributed null-controllability of some classes of linear parabolic systems. On the other hand, while for scalar problems the boundary controllability is known to be equivalent to the distributed controllability, it has been proved in [FCGBdT10] that this is no more the case for systems. This reveals that the controllability of systems is much more subtle. In [AKBGBdT12], it is even showed that a minimal time of control can appear if the diffusion is different on each equation, which is quite surprising for a system possessing an infinite speed of propagation. It is important to emphasize that the previous quoted results concerning the boundary controllability were established in space dimension one. They used the moment method, generalizing the works of [FR71,FR75] concerning the boundary controllability of the one-dimensional scalar heat equation. We refer to [AKBGBdT11b] for more details and a survey on the controllability of parabolic systems.In higher space dimension the boundary controllability of parabolic systems remains widely open and it is the main purpose of this article to give some partial answers. To our knowledge, the only results on this issue are the one of [ABL12] and [AB12]. Let us also mention [Oli13] for related questions for the approximate controllability problem. In [ABL12, AB12] the results for parabolic systems are deduced from the study of the boundary control problem of two coupled wave equations using transmutation techniques. As a result there are some geometric constraints on the control domain. We will see that this restriction is not necessary.In the present work, we focus on the boundary null-controllability of the following n coupled parabolic equations by m controls in dimension N > 1

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“…To our knowledge, the only results on this issue are the one of [ABL12] and [AB12]. Let us also mention [Oli13] for related questions for the approximate controllability problem.…”

mentioning

confidence: 99%

Sharp Estimates of the One-Dimensional Boundary Control Cost for Parabolic Systems and Application to the $N$-Dimensional Boundary Null Controllability in Cylindrical Domains

Benabdallah¹,

Boyer²,

González-Burgos³

et al. 2014

SIAM J. Control Optim.

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“…In the present work, we answer these two questions for hyperbolic problems improving the results of [2,3], and then deduce a (partial) solution to the two open questions raised above. Indeed, we prove that Systems (2)-(3) are null-controllable (in appropriate spaces) as soon as {p > 0} and {b > 0} both satisfy the Geometric Control Condition (recalled below) and √ δ p L ∞ (Ω) satisfies a smallness assumption.…”

Section: Introductionmentioning

confidence: 75%

“…In this section, we describe the abstract setting (already used in [3]) in which we prove Theorem 2.1 for Systems (3)- (5), and define the appropriate spaces and operators. Let H be a Hilbert space and (A, D(A)) a selfadjoint positive operator on H with compact resolvent.…”

Section: Abstract Setting and Ingredients Of Proofmentioning

confidence: 99%

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Indirect controllability of locally coupled systems under geometric conditions

Alabau‐Boussouira¹,

Léautaud

2011

Comptes Rendus. Mathématique

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We consider systems of two wave/heat/Schrödinger-type equations coupled by a zero order term, only one of them being controlled. We prove an internal and a boundary null-controllability result in any space dimension, provided that both the coupling and the control regions satisfy the Geometric Control Condition. This includes several examples in which these two regions have an empty intersection. AbstractContrôlabilité indirecte de systèmes localement couplés sous des conditions géométriques. On s'intéresseà des systèmes constitués de deuxéquations d'ondes, de la chaleur ou de Schrödinger, couplées par un terme d'ordre zéro, et dont seulement l'une est controlée. En supposant que les zones de couplage et de contrôle satisfont toutes deux la Condition Géométrique de Contrôle, on montre un résultat de contrôle interne et frontière en dimension quelconque d'espace. Ceci fournit de nombreux exemples pour lesquels ces deux régions ne s'intersectent pas.

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“…The stability of various specific infinite-dimensional coupled systems has been investigated in a series of papers by J.M. Wang and co-authors, see [16], [45], [46], also by F. Alabau-Boussouira and co-authors [3], [4], S. Hansen and coauthors [17], [18], and our paper [56]. Many further references on coupled systems can be found in these works.…”

Section: Introductionmentioning

confidence: 97%

Stability Properties of Coupled Impedance Passive LTI Systems

Zhao

2017

IEEE Trans. Automat. Contr.

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Abstract-We study the stability of the feedback interconnection of two impedance passive linear time-invariant systems, of which one is finite-dimensional. The closed-loop system is well known to be impedance passive, but no stability properties follow from this alone. We are interested in two main issues: (1) the strong stability of the operator semigroup associated with the closed-loop system, (2) the input-output stability (meaning transfer function in H ∞ ) of the closed-loop system. Our results are illustrated with the system obtained from the non-uniform SCOLE (NASA Spacecraft Control Laboratory Experiment) model representing a vertical beam clamped at the bottom, with a rigid body having a large mass on top, connected with a trolley mounted on top of the rigid body, via a spring and a damper. Such an arrangement called a tuned mass damper (TMD), is used to stabilize tall buildings. We show that the SCOLE-TMD system is strongly stable on the energy state space and that the system is input-output stable from the horizontal force input to the horizontal velocity output.

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Indirect stabilization of locally coupled wave-type systems

Cited by 57 publications

References 27 publications

Sharp Estimates of the One-Dimensional Boundary Control Cost for Parabolic Systems and Application to the $N$-Dimensional Boundary Null Controllability in Cylindrical Domains

Sharp Estimates of the One-Dimensional Boundary Control Cost for Parabolic Systems and Application to the $N$-Dimensional Boundary Null Controllability in Cylindrical Domains

Indirect controllability of locally coupled systems under geometric conditions

Stability Properties of Coupled Impedance Passive LTI Systems

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