Controllability of the Parabolic System Via Bilinear Control

Tsouli, A.; Boutoulout, Ali

doi:10.1007/s10883-014-9247-2

Cited by 7 publications

(5 citation statements)

References 16 publications

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“…Then using (19) and (25), we get T 0 | B S(t)z, S(t)z |dt = 0, so B S(t)z, S(t)z = 0, ∀t ≥ 0, which gives, by virtue of (3) that z = 0. Hence y u (t) 0, as t → +∞, and using the fact that dim H u < +∞, we deduce that y u (t) → 0 as t → +∞.…”

Section: Theorem 21 1 Let a Generate A Linear C 0 -Semigroup S(t) Omentioning

confidence: 99%

“…Bilinear systems have been considered since the early 1960s as a gateway between linear and nonlinear systems that are defined to be linear in both state and control when considered independently, with the nonlinearity (or bilinearity) arising from coupled terms involving products of system state and control (see [8,19]). By formulating the model appropriately, the bilinear term could also be represented by products of system output and control input, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition

Tsouli

Boutoulout

2015

Rend. Circ. Mat. Palermo

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In this paper, we shall study the stability for distributed bilinear systems on a Hilbert state space that can be decomposed in two subspaces: unstable finite-dimensional and stable infinite-dimensional with respect to the evolution generator. Then, we shall show under a weaker observability assumption that stabilizing such a system with a feedback control of the form p r (t) = − y(t) −r y(t), By(t) for r < 2, reduces stabilizing only its projection on the finite-dimension subspace which make the whole system stable. To this end, we shall propose a new family of continuous feedback controls that guarantee the uniform stabilizability with an explicit optimal decay rate estimate of the stabilized state.Two illustrating examples and simulations are provided.

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Section: Theorem 21 1 Let a Generate A Linear C 0 -Semigroup S(t) Omentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition

Tsouli

Boutoulout

2015

Rend. Circ. Mat. Palermo

Self Cite

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“…The equation (1.1) has been studied in [7], in the whole space and with controls localized everywhere in space and time. Concerning bilinear control when the bilinear term div(uy) is replaced by uy with u ∈ L ∞ ((0, T ) × Ω), we refer to [8,9,27,28,26,29,34,40,45,46].…”

Section: Introductionmentioning

confidence: 99%

Bilinear local controllability to the trajectories of the Fokker–Planck equation with a localized control

Duprez¹,

Lissy²

2022

Annales De l'Institut Fourier

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This work is devoted to the control of the Fokker-Planck equation, posed on a smooth bounded domain of R d , with a localized drift force. We prove that this equation is locally controllable to regular nonzero trajectories. Moreover, under some conditions, we explain how to reduce the number of controls around the reference control. The results are obtained thanks to a standard linearization method and the fictitious control method. The main novelties are twofold. First, the algebraic solvability is performed and used directly on the adjoint problem. We then prove a new Carleman inequality for the heat equation with a space-time varying first-order term: the right-hand side is the gradient of the solution localized on an open subset. We finally give an example of regular trajectory around which the Fokker-Planck equation is not controllable with a reduced number of controls, to highlight that our conditions are relevant.Résumé. -Ce travail est consacré au contrôle de l'équation de Fokker-Planck, posée sur un domaine borné régulier de R d , avec un terme de dérive localisé. Nous démontrons que cette équation est localement contrôlable aux trajectoires régulières non nulles. De plus, sous certaines conditions, nous expliquons comment réduire le nombre de contrôles autour du contrôle de référence. Les résultats sont obtenus à l'aide d'une méthode de linéarisation standard et la méthode de contrôle fictif. Les principales nouveautés sont les suivantes. Premièrement, la résolubilité algébrique est effectuée et utilisée directement sur le problème adjoint. Deuxièmement, nous démontrons une nouvelle inégalité de Carleman pour l'équation de la chaleur avec terme du premier ordre dépendant du temps et de l'espace : le membre de droite est le gradient de la solution localisée sur un sous-ouvert. Pour finir, nous donnons un exemple de trajectoire régulière autour de laquelle l'équation de Fokker-Planck n'est pas contrôlable ave un nombre réduit de contrôles, pour souligner que nos conditions sont pertinentes.

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“…Equation (1.1) has been studied in [8], in the whole space and with controls localized everywhere in space and time. Concerning bilinear control in the case where the bilinear term div(uy) is replaced by uy with u ∈ L ∞ ((0, T ) × Ω), we refer to [9,10,27,28,26,29,34,40,44,45].…”

Section: Introductionmentioning

confidence: 99%

Bilinear local controllability to the trajectories of the Fokker-Planck equation with a localized control

Duprez,

Lissy

2019

Preprint

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This work is devoted to the control of the Fokker-Planck equation, posed on a bounded domain of R d (d 1). More precisely, the control is the drift force, localized on a small open subset. We prove that this system is locally controllable to regular nonzero trajectories. Moreover, under some conditions on the reference control, we explain how to reduce the number of controls around the reference control. The results are obtained thanks to a linearization method based on a standard inverse mapping procedure and the fictitious control method. The main novelties of the present article are twofold. Firstly, we propose an alternative strategy to the standard fictitious control method: the algebraic solvability is performed and used directly on the adjoint problem. Secondly, we prove a new Carleman inequality for the heat equation with one order space-varying coefficients: the right-hand side is the gradient of the solution localized on a subset (rather than the solution itself), and the left-hand side can contain arbitrary high derivatives of the solution.

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Controllability of the Parabolic System Via Bilinear Control

Abstract: The aim of this paper is to prove approximate and exact controllability for a multidimensional heat equation governed by bilinear control with nonhomogeneous boundary conditions in a bounded domain. For this purpose, an explicit control strategy is constructed.

Cited by 7 publications

References 16 publications

Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition

Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition

Bilinear local controllability to the trajectories of the Fokker–Planck equation with a localized control

Bilinear local controllability to the trajectories of the Fokker-Planck equation with a localized control

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