Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form

Cannarsa, Piermarco; Fragnelli, Genni; Rocchetti, Dario

doi:10.1007/s00028-008-0353-34

Cited by 76 publications

(86 citation statements)

References 26 publications

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“…Then, null controllability holds if and only if γ ∈ (0, 1) [22,23], while, for γ ≥ 1, the best result one can obtain is the so called regional null controllability [21], which consists in controlling the solution within the domain of inuence of the control. Several extensions of the above results are available in one space dimension, see [2,49] for equations in divergence form, [20,19] …”

Section: Boundary-degenerate Parabolic Equationsmentioning

confidence: 83%

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: Boundary-degenerate Parabolic Equationsmentioning

confidence: 83%

Degenerate parabolic operators of Kolmogorov type with a geometric control condition

et al. 2015

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“…Then, null controllability holds if and only if γ ∈ (0, 1) (see [13,14]), while, for γ ≥ 1, the best result one can show is regional null controllability(see [12]), which consists in controlling the solution within the domain of inuence of the control. Several extensions of the above results are available in one space dimension, see [1,34] for equations in divergence form, [11,10] for nondivergence form operators, and [9,24] for cascade systems.…”

Section: Boundary-degenerate Parabolic Equationsmentioning

confidence: 93%

Null controllability of Kolmogorov-type equations

Beauchard

2013

Math. Control Signals Syst.

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International audienceWe study the null controllability of Kolmogorov-type equations in a rectangle, under an additive control supported in an open subset of the rectangle. These equations couple a diffusion in variable v with a transport in variable x at speed v^a. For a=1, with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support.This improves the previous result [5], in which the control support was a horizontal strip. With Dirichlet boundary conditions and a horizontal strip as control support, we prove that null controllability holds in any positive time if a = 1, or if a= 2 and the control support contains the segment {v = 0}, and only in large time if a = 2 and the control support does not contain the segment {v = 0}. Our approach, inspired from [7, 31], is based on 2 key ingredients: the observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the frequency, and the explicit exponential decay rate of these Fourier components

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“…Several extensions of the above results are available in one space dimension, see [1,37] for equations in divergence form, [11,10] for nondivergence form operators, and [9,25] for cascade systems. Fewer results are available for multidimensional problems, mainly in the case of two dimensional parabolic operators which simply degenerate in the normal direction to the boundary of the space domain, see [15].…”

Section: Xxxiv-5mentioning

confidence: 88%

“…Let g be the solution of (7)- (11). Then, g can be represented as in (17), and we emphasize that, for a.e.…”

Section: Strategy For the Proofmentioning

confidence: 99%

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Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type

Beauchard

2014

Séminaire Laurent Schwartz — EDP Et Applications

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Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form

Cited by 76 publications

References 26 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 Kolmogorov-type equations

Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type

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