Null controllability of degenerate parabolic operators with  drift

Cannarsa, Piermarco; Fragnelli, Genni; Rocchetti, Dario

doi:10.3934/nhm.2007.2.695

Cited by 70 publications

(63 citation statements)

References 24 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: 84%

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: 84%

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: 99%

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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“…We should also point out that nondivergence operators in one dimension (also degenerate ones) and their probabilistic interpretation are studied by Mandl [24]. An application to mathematical finance is contained in Cannarsa et al [9].…”

Section: Introductionmentioning

confidence: 99%

Dirichlet regularity and degenerate diffusion

Arendt¹,

Chovanec²

2010

Trans. Amer. Math. Soc.

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Abstract. Let Ω ⊂ R N be an open and bounded set and let m : Ω → (0, ∞) be measurable and locally bounded. We study a natural realization of the operator m in C 0 (Ω) := u ∈ C(Ω) : u |∂Ω = 0 . If Ω is Dirichlet regular, then the operator generates a positive contraction semigroup on C 0 (Ω) wheneverdoes not go fast enough to 0 as x → ∂Ω, then Dirichlet regularity is necessary. However, if |m(x)| ≤ c·dist(x, ∂Ω) 2 , then we show that m 0 generates a semigroup on C 0 (Ω) without any regularity assumptions on Ω. We show that the condition for degeneration of m near the boundary is optimal.

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Null controllability of degenerate parabolic operators with drift

Cited by 70 publications

References 24 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

Dirichlet regularity and degenerate diffusion

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