Unique continuation property near a corner and its fluid-structure controllability consequences

Osses, Axel; Puel, Jean-Pierre

doi:10.1051/cocv:2008024

Cited by 3 publications

(3 citation statements)

References 10 publications

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“…We show that the stabilizability of system (1.2) reduces to proving an approximate controllability result for a projected system. Such an approximate controllability result can be deduced from [18] in the case of a rectangular domain (see also [19,20] for supplementary approximate controllability results).…”

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confidence: 99%

Feedback Stabilization of a Fluid-Structure Model

Raymond¹

2010

SIAM J. Control Optim.

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We study a system coupling the incompressible Navier-Stokes equations in a 2D rectangular type domain with a damped Euler-Bernoulli beam equation, where the beam is a part of the upper boundary of the domain occupied by the fluid. Due to the deformation of the beam the fluid domain depends on time. We prove that this system is exponentially stabilizable, locally about the null solution, with any prescribed decay rate, by a feedback control corresponding to a force term in the beam equation. The feedback is determined, via a Riccati equation, by solving an infinite time horizon control problem for the linearized model. A crucial step in this analysis consists in showing that this linearized system can be rewritten thanks to an analytic semigroup of which the infinitesimal generator has a compact resolvent.

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confidence: 99%

Feedback Stabilization of a Fluid-Structure Model

Raymond¹

2010

SIAM J. Control Optim.

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“…We have also to mention the works [26] which studies a related unique continuation problem for Stokes equations and [30], where a unique continuation result is proved in the case of a channel with Dirichlet boundary conditions on the whole boundary. See also [31], [32] for more refined results of the same type.…”

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confidence: 90%

“…32) where β solves (4.1) with α solving (3.9) with control g(t,x 1 ) = ag s ref (t) sin(x 1 ) + bg c ref (t) cos(x 1 ), and q ref denotes the pressure obtained by solving (4.30) with source term…”

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confidence: 99%

Open loop stabilization of incompressible Navier–Stokes equations in a 2d channel using power series expansion

Chowdhury

Ervedoza

2019

Journal de Mathématiques Pures et Appliquées

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In this article, we discuss the stabilization of incompressible Navier-Stokes equations in a 2d channel around a fluid at rest when the control acts only on the normal component of the upper boundary. In this case, the linearized equations are not controllable nor stabilizable at an exponential rate higher than νπ 2 /L 2 , when the channel is of width L and of length 2π and ν denotes the viscosity parameter. Our main result allows to go above this threshold and reach any exponential decay rate by using the non-linear term to control the directions which are not controllable for the linearized equations. Our approach therefore relies on writing the controlled trajectory as an expansion of order two taking the form εα + ε 2 β for some ε > 0 small enough. This method is inspired by the previous work [18] by J.-M. Coron and E. Crépeau on the controllability of the Korteweg de Vries equations.In this section, we recall some basic facts on the linearized equation (1.4) and give a modal description adapted to our setting.

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Feedback Stabilization of a Two-Dimensional Fluid-Structure Interaction System with Mixed Boundary Conditions

Fournié¹,

Ndiaye²,

Raymond³

2019

SIAM J. Control Optim.

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We study the stabilization of a fluid-structure interaction system around an unstable stationary solution. The system consists of coupling the incompressible Navier-Stokes equations, in a two dimensional polygonal domain with mixed boundary conditions, and a damped Euler-Bernoulli beam equations located at the boundary of the fluid domain. The control acts only in the beam equations. The feedback is determined by stabilizing the projection of the linearized model onto a finite dimensional invariant subspace. Here we have resolved two important challenges for applications in this field. One is the fact that we prove a stabilization result around a non zero stationary solution, which is new for such fluid-structure interaction systems. The other one is that the feedback laws that we determine do not depend on the Leray projector used to get rid of the algebraic constraints of partial differential equations. This is essential for numerical aspects.

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Unique continuation property near a corner and its fluid-structure controllability consequences

Cited by 3 publications

References 10 publications

Feedback Stabilization of a Fluid-Structure Model

Feedback Stabilization of a Fluid-Structure Model

Open loop stabilization of incompressible Navier–Stokes equations in a 2d channel using power series expansion

Feedback Stabilization of a Two-Dimensional Fluid-Structure Interaction System with Mixed Boundary Conditions

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