Local Boundary Feedback Stabilization of the Navier-Stokes Equations

Raymond, Jean‐Pierre

doi:10.22323/1.018.0003

Cited by 11 publications

(14 citation statements)

References 17 publications

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“…Here, motivated by stabilization problems [22,23], we would like to find optimal regularity results for the solution to the Stokes and the Oseen equations when g belongs to the space We are also interested in finding a sufficient condition on g so that a weak solution to Eq. (1.1) exists in the case where κ = 1.…”

Section: Introductionmentioning

confidence: 99%

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

Raymond

2007

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by P u, the projection of the solution on the Stokes space-the space of divergence free vector fields with a normal trace equal to zero-and the second one is a quasi-stationary elliptic equation satisfied by (I − P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for P u and (I − P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier-Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.

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Section: Introductionmentioning

confidence: 99%

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

Raymond

2007

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…Finally mention that in the three dimensional case, the stabilization of the nonlinear problem cannot be treated with the same tools and requires a more delicate analysis [31,32].…”

Section: An Algebraic Riccati Equation Is Stated Only In D((a π )mentioning

confidence: 99%

Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations

Raymond¹

2006

SIAM J. Control Optim.

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154

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Abstract. We study the exponential stabilization of the linearized Navier-Stokes equations around an unstable stationary solution, by means of a feedback boundary control, in dimension 2 or 3. The feedback law is determined by solving a Linear-Quadratic control problem. We do not assume that the normal component of the control is equal to zero. In that case the state equation, satisfied by the velocity field y, is decoupled into an evolution equation satisfied by P y, where P is the so-called Helmholtz projection operator, and a quasi-stationary elliptic equation satisfied by (I − P )y. Using this decomposition we show that the feedback law can be expressed only in function of P y. In the two dimensional case we show that the linear feedback law provides a local exponential stabilization of the Navier-Stokes equations.Key words. Dirichlet control, feedback control, stabilization, Navier-Stokes equations, Oseen equations, Riccati equation AMS subject classifications. 93B52, 93C20, 93D15, 35Q30, 76D55, 76D05, 76D071. Setting of the problem. Let Ω be a bounded and connected domain in R 2 or R 3 with a regular boundary Γ, ν > 0, and consider a couple (w, χ) -a velocity field and a pressure -solution to the stationary Navier-Stokes equations in Ω:We assume that w is regular and is an unstable solution of the instationary Navier-Stokes equations. The purpose of this paper is to determine a Dirichlet boundary control u, in feedback form, localized in a part of the boundary Γ, so that the corresponding controlled system:be stable for initial values y 0 small enough in an appropriate space X(Ω). In this setting, Q ∞ = Ω×(0, ∞),in Ω, y · n = 0 on Γ , and the operator M is a restriction operator precisely defined in section 2. If we set (z, q) = (w + y, χ + p) and if u = 0, we see that (z, q) is the solution to the Navier-Stokes equationsThus y 0 is a perturbation of the stationary solution w.To study the local feedback stabilization of system (1.2), we first study the feedback stabilization of the corresponding linearized system(1.3)To stabilize this system we can look for a control u belonging either to L 2 (0, ∞; V 0 (Γ)) or to L 2 (0, ∞; V 0 n (Γ)), where V 0 (Γ) = y ∈ L 2 (Γ) | y · n, 1 H −1/2 (Γ),H 1/2 (Γ) = 0 and V 0 n (Γ) = y ∈ L 2 (Γ) | y · n = 0 on Γ .

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“…The feedback stabilization of laminar flows based on controllers designed to attenuate deviations from a stationary working point has got a substantial theoretical foundation, see Raymond (2005Raymond ( , 2006. The basic idea is to linearize the model about an unstable target steady-state and to define a controller based on the linearization that is capable to damp small perturbations of the target state also in the actual nonlinear model.…”

Section: Introductionmentioning

confidence: 99%

Robust Stabilization of Laminar Flows in Varying Flow Regimes

Benner

Heiland

2016

IFAC-PapersOnLine

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The stabilization of laminar flows on the base of linearizations and feedback controllers has been the subject of many recent theoretical and computational studies. However, the applicability of the standard approaches is limited due to the inherent fragility of observer based controllers with respect to arbitrary small changes in the system. We show that a slight variation in the Reynolds number of a flow setup amounts to a coprime factor perturbation in the associated linear transfer function. Based on these findings, we argue that known concepts from robust control can be exploited to come up with an output feedback law that can stabilize the cylinder wake over the transition period from a stable to a stabilized regime.

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Local Boundary Feedback Stabilization of the Navier-Stokes Equations

Cited by 11 publications

References 17 publications

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations

Robust Stabilization of Laminar Flows in Varying Flow Regimes

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