Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

Beauchard, Karine; Egidi, Michela; Pravda-Starov, Karel

doi:10.5802/crmath.79

Cited by 15 publications

(35 citation statements)

References 35 publications

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“…The strategy consists in applying this observability estimate for well-chosen functions g ∈ L 2 (R n ). This approach has especially been used in the works [2,3,4,9] in which stabilization or null-controllability issues are studied for fractional heat equations or evolution equations associated with (non)-autonomous Ornstein-Uhlenbeck operators posed on the whole space R n . Fixing x 0 ∈ R n and considering ξ 0 ∈ R n and l ≫ 1 whose values will be adjusted later, we consider the Gaussian function g l,ξ 0 defined by…”

Section: (Rapid) Stabilization Of Diffusive Equationsmentioning

confidence: 99%

“…Other classes of degenerate parabolic equations of hypoelliptic type, as evolution equations associated with accretive quadratic operators or (non-autonomous) Ornstein-Uhlenbeck operators, were also proven to be null-controllable from thick control supports, see e.g. [2,4,5]. In this work, we prove that a very general class of equations of the form (E F ) are approximately null-controllable with uniform cost from the support control ω ⊂ R n if and only if ω is a thick set.…”

Section: Introductionmentioning

confidence: 99%

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Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

Alphonse¹,

Martin²

2021

Preprint

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We prove that the thickness is a necessary and sufficient geometric condition that ensures the (rapid) stabilization or the approximate null-controllability with uniform cost of a large class of evolution equations posed on the whole space R n and associated with operators of the form F (|Dx|), the function F : [0, +∞) → R being bounded below and continuous. We also provide explicit feedbacks and constants associated with these stabilization properties. Our results apply in particular for the half heat equation associated with the function F (t) = t, for which null-controllability is known to fail from thick control supports. More generally, the notion of thickness was known to be a necessary and sufficient condition for the null-controllability of the fractional heat equations associated with the functions F (t) = t 2s in the case s > 1, and that this null-controllability property from thick control supports does not hold when 0 < s ≤ 1.

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Section: (Rapid) Stabilization Of Diffusive Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

Alphonse¹,

Martin²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The strategy consists in applying this observability estimate for well-chosen functions g ∈ L 2 (R n ). This approach has especially been used in the works [2,3,4,9], in which exponential stabilization or null-controllability issues are studied for fractional heat equations or evolution equations associated with (non)-autonomous Ornstein-Uhlenbeck operators posed on the whole space R n . Fixing x 0 ∈ R n and considering ξ 0 ∈ R n together with l ≫ 1 whose values will be adjusted later, we consider the Gaussian function g l,ξ 0 defined by…”

Section: (Rapid) Stabilization Of Diffusive Equationsmentioning

confidence: 99%

“…Other classes of degenerate parabolic equations of hypoelliptic type, as evolution equations associated with accretive quadratic operators or (non-autonomous) Ornstein-Uhlenbeck operators, were also proven to be null-controllable from thick control supports, see e.g. [2,4,5]. In this work, we prove that a very general class of equations of the form (E F ) is approximately null-controllable with uniform cost from the control support ω ⊂ R n if and only if ω is a thick set.…”

Section: Introductionmentioning

confidence: 99%

Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

Alphonse

Martin

2022

ESAIM: COCV

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We prove that the thickness property is a necessary and sufficient geometric condition that ensures the (rapid) stabilization or the approximate null-controllability with uniform cost of a large class of evolution equations posed on the whole space $\mathbb R^n$. These equations are associated with operators of the form $F(\vert D_x\vert)$, the function $F:[0,+\infty)\rightarrow\mathbb R$ being continuous and bounded from below. We also provide explicit feedbacks and constants associated with these stabilization properties. The notion of thickness is known to be a necessary and sufficient condition for the exact null-controllability of the fractional heat equations associated with the functions $F(t) = t^{2s}$ in the case $s>1/2$. Our results apply in particular for this class of equations, but also for the half heat equation associated with the function $F(t) = t$, which is the most diffusive fractional heat equation for which exact null-controllability is known to fail from general thick control supports.

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“…The study of the (rapid) stabilization and the (approximate) null-controllability of parabolic equations [4,12,14,16,20,25,28] or degenerate parabolic equations of hypoelliptic type [3,7,8,9,11] posed on R n and taking the following form (E P ) ∂ t f (t, x) + P f (t, x) = h(t, x)½ ω (x), (t, x) ∈ (0, +∞) × R n ,…”

Section: Introductionmentioning

confidence: 99%

Approximate null-controllability with uniform cost for the hypoelliptic Ornstein-Uhlenbeck equations

Alphonse,

Martin

2022

Preprint

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We prove that the approximate null-controllability with uniform cost of the hypoelliptic Ornstein-Uhlenbeck equations posed on R n is characterized by an integral thickness geometric condition on the control supports. We also provide associated quantitative weak observability estimates. This result for the hypoelliptic Ornstein-Uhlenbeck equations is deduced from the same study for a large class of non-autonomous elliptic equations from moving control supports. We generalize in particular results known for parabolic equations posed on R n , for which the approximate null-controllability with uniform cost is ensured by the notion of thickness, which is stronger that the integral thickness condition considered in the present work. Examples of those parabolic equations are the fractional heat equations associated with the operator (−∆) s , in the regime s ≥ 1/2. Our strategy also allows to characterize the approximate null-controllability with uniform cost from moving control supports for this class of fractional heat equations.

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Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

Abstract: Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

Cited by 15 publications

References 35 publications

Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

Stabilization and approximate null-controllability for a large class of diffusive equations from thick control supports

Approximate null-controllability with uniform cost for the hypoelliptic Ornstein-Uhlenbeck equations

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