Null controllability of a thermoelastic plate

Benabdallah, Assia; Naso, Maria Grazia

doi:10.1155/s108533750220408x

Cited by 43 publications

(49 citation statements)

References 13 publications

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“…The method introduced in [LR95] was further extended to address thermoelasticity [LZ98], thermoelastic plates [BN02], semigroups generated by fractional orders of elliptic operators [Mil06]. It has also been used to prove null controllability results in the case of non smooth coefficients [BDL07b,LR10a].…”

Section: Introductionmentioning

confidence: 99%

On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

2011

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Section: Introductionmentioning

confidence: 99%

On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

2011

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“…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.…”

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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“…Finally we quote the following papers concerned with the controllability of the Gurtin-Pipkin equation: [1,3,10,12] …”

Section: Pos(cstna2005)015mentioning

confidence: 99%

Controllability of the Gurtin-Pipkin equation

Pandolfi¹

2006

Proceedings of Control Systems: Theory, Numerics and Applications — PoS(CSTNA2005)

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The Gurtin-Pipkin equations models the evolution of thermal phenomena when the matherial keeps memory of the past. It is well known that the Gurtin-Pipkin equation has a hyperbolic type behavior so that it is natural to expect a property of exact observability for its solutions. In this talk we consider the Gurtin-Pipkin equation with control in the Diriclet boundary condition and we prove exact controllability as a consequence of the known exact controllability of the wave equation.

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Null controllability of a thermoelastic plate

Cited by 43 publications

References 13 publications

On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

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

Controllability of the Gurtin-Pipkin equation

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