A Family of Stabilization Problems for the Oseen Equations
The feedback stabilization of the Navier-Stokes equations around an unstable stationary solution is related to the feedback stabilization of the Oseen equations (the linearized Navier-Stokes equations about the unstable stationary solution). In this paper
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Family of Stabilization Problems for the Oseen Equations Jean-Pierre Raymond Abstract. The feedback stabilization of the Navier-Stokes equations around an unstable stationary solution is related to the feedback stabilization of the Oseen equations (the linearized Navier-Stokes equations about the unstable stationary solution). In this paper we investigate the regularizing properties of feedback operators corresponding to a family of optimal control problems for the Oseen equations.
1. Introduction Let Ω be a bounded and connected domain in R3 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: −ν∆w + (w · ∇)w + ∇χ = f
and div w = 0 in Ω,
w = u∞ s
on Γ.
We assume that w is regular and is an unstable solution of the instationary NavierStokes equations. The local feedback boundary stabilization of the Navier-Stokes equations consists in finding a Dirichlet boundary control u, in feedback form, localized in a part of the boundary Γ, so that the corresponding control system: ∂y − ν∆y + (y · ∇)w + (w · ∇)y + (y · ∇)y + ∇p = 0 in Q∞ , ∂t div y = 0 in Q∞ , y = M u on Σ∞ , y(0) = y0 in Ω,
(1)
be stable for initial values y0 small enough in an appropriate space X(Ω). In2this setting, Q∞ = Ω × (0, ∞), Σ∞ = Γ × (0, ∞), X(Ω) is a subspace of Vn0 (Ω) = y ∈ 3 L2 (Ω) | div y = 0 in Ω, y · n = 0 on Γ , n is the unit normal to Γ outward Ω, y0 ∈ X(Ω), and the operator M is a restriction operator defined in Section 2.
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J.-P. Raymond
The feedback stabilization of equations (1) is closely related to the feedback stabilization of the Oseen equations: ∂y − ν∆y + (w · ∇)y + (y · ∇)w + ∇p = 0, in Q∞ , ∂t (2) div y = 0 in Q∞ , y = M u on Σ∞ , y(0) = y0 in Ω. One way to determine a feedback control law able to stabilize system (2) consists in solving an optimal control problem with an infinite time horizon. Once the functional of the control problem is defined the feedback law can be determined by calculating the solution to the corresponding algebraic Riccati equation (if it exists and if it admits a unique solution). Changing the functional, we change the feedback operator. These functionals are generally of the form 1 ∞ 1 ∞ 2 |Cy| dxdt + |u|2 dxdt, J(y, u) = 2 0 2 0 Ω Γ where C, the 2 observation operator, may be a bounded or an unbounded operator 3 0 2 in V (Ω) = y ∈ L (Ω) | div y = 0 in Ω, Γ y · n = 0 on Γ . When C is unbounded let us denote by D(C) its domain in V0 (Ω). Let us consider the optimal control problem 2 inf J(y, u) | (y, u) satisfies (2), u ∈ L2 (0, ∞; V0 (Γ)), (Q) 3 y ∈ L2 (0, ∞; D(C)) , where
2 3 2 V (Γ) = y ∈ L (Γ) | y·n= 0 . 0
Γ
In two dimension [15], we have determined a feedback law by choosing C = I. For such a choice, we have shown that the optimal solution (yy0 , uy0 ) to (Q) obeys a feedback formula of the form −1 M B ∗ ΠP yy0 (t), uy0 (t) = −RA
where P is the so-called Helmholtz or Leray3projector in L2 (Ω) onto Vn0 (Ω) = 2 y ∈ L2 (Ω) | div y = 0 in Ω, y · n = 0 on Γ , Π is the solution to
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