Null controllability for the heat equation with singular inverse-square potentials

Vancostenoble, Judith; Zuazua, Enrique

doi:10.1016/j.jfa.2007.12.015

Cited by 70 publications

(88 citation statements)

References 19 publications

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“…The null and approximate controllabilities of the heat equation are essentially well understood subjects for both linear and semilinear equations, for bounded or unbounded domains [3,27,30,32,33,34,37,44,46,48,51,52,62,63] and also with discontinuous [28,12,13,57] or singular [61,29] coecients.…”

Section: Null Controllability Of the Heat Equationmentioning

confidence: 99%

Degenerate parabolic operators of Kolmogorov type with a geometric control condition

et al. 2015

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We consider Kolmogorov-type equations on a rectangle domain (x, v) ∈ Ω = T × (−1, 1), that combine diusion in variable v and transport in variable x at speed v γ , γ ∈ N * , with Dirichlet boundary conditions in v. We study the null controllability of this equation with a distributed control as source term, localized on a subset ω of Ω.In dimension one, when the control acts on a horizontal strip ω = T × (a, b) with 0 < a < b < 1, then the system is null controllable in any time T > 0 when γ = 1, and only in large time T > Tmin > 0 when γ = 2 (see [10]). In this article, we prove that, when γ > 3, the system is not null controllable (whatever T is) in this conguration. This is due to the diusion weakening produced by the rst order term.When the control acts on a vertical strip ω = ω1 × (−1, 1) with ω1 ⊂ T, we investigate the null controllability on a toy model, where (∂x, x ∈ T) is replaced by1/2 , x ∈ Ω1), and Ω1 is an open subset of R N . As the original system, this toy model satises the controllability properties listed above. We prove that, for γ = 1, 2 and for appropriate domains (Ω1, ω1), then null controllability does not hold (whatever T > 0 is), when the control acts on a vertical strip ω = ω1 × (−1, 1) with ω1 ⊂ Ω1. Thus, a geometric control condition is required for the null controllability of this toy model. This indicates that a geometric control condition may be necessary for the original model too.

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Section: Null Controllability Of the Heat Equationmentioning

confidence: 99%

Degenerate parabolic operators of Kolmogorov type with a geometric control condition

et al. 2015

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“…[17], [7], [8], [41]) or singular ( [42] and [19]) coefficients. In particular, the heat equation on a smooth bounded domain…”

Section: Null Controllability Of the Heat Equationmentioning

confidence: 99%

Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type

Beauchard

2014

Séminaire Laurent Schwartz — EDP Et Applications

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“…4. In the following, we assume that (1.4) holds for the continuous system. Such results have been proved, often by means of Carleman estimates, for various models including the heat equation [10,12,16], the Stokes equations [9], and some other singular models such as [3,7,20,24].…”

Section: Introductionmentioning

confidence: 99%

On the observability of abstract time-discrete linear parabolic equations

Ervedoza

Valein

2009

Rev Mat Complut

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This article aims at analyzing the observability properties of time-discrete approximation schemes of abstract parabolic equationsż + Az = 0, where A is a selfadjoint positive definite operator with dense domain and compact resolvent. We analyze the observability properties of these diffusive systems for an observation operator B ∈ L(D(A ν ), Y ) with ν < 1/2. Assuming that the continuous system is observable, we prove uniform observability results for suitable time-discretization schemes within the class of conveniently filtered data. We also propose a HUM type algorithm to compute discrete approximations of the exact controls. Our approach also applies to sequences of operators which are uniformly observable. In particular, our results can be combined with the existing ones on the observability of space semi-discrete systems, yielding observability properties for fully discrete approximation schemes. Mathematics Subject Classification (2000)35K05 · 93B05 · 93B07 · 93B40 · 93C55

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Null controllability for the heat equation with singular inverse-square potentials

Cited by 70 publications

References 19 publications

Degenerate parabolic operators of Kolmogorov type with a geometric control condition

Degenerate parabolic operators of Kolmogorov type with a geometric control condition

Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type

On the observability of abstract time-discrete linear parabolic equations

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