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 .

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The control is the trace of the state on the boundary of the domain, which is assumed to be a convex, polygonal, open set in R 2 . Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the error estimates are of order O(h 1−1/p ) for some p > 2, which is consistent with the W 1−1/p,p (Γ)-regularity of the optimal control. 1. Introduction. In this paper we study an optimal control problem governed by a semilinear elliptic equation. The control is the Dirichlet datum on the boundary of the domain. Bound constraints are imposed on the control. The cost functional involves the control in a quadratic way and the state in a general way. The goal is to derive error estimates for the discretization of the control problem.There are not many papers devoted to the derivation of error estimates for the discretization of control problems governed by partial differential equations; see the pioneering works by Falk [19] and Geveci [21]. However, recently some papers have appeared, providing new methods and ideas. Arada, Casas, and Tröltzsch [1] derived error estimates for the controls in the L ∞ and L 2 norms for distributed control problems. Similar results for an analogous problem, but also including integral state constraints, were obtained by Casas [8]. The case of a Neumann boundary control problem has been studied by Casas, Mateos, and Tröltzsch [11]. The novelty of our paper with respect to the previous ones is twofold. First, here we deal with a Dirichlet problem, the control being the value of the state on the boundary. Second, we consider piecewise linear continuous functions to approximate the optimal control, which is necessary because of the Dirichlet nature of the control, but it introduces some new difficulties. In the previous papers the controls were always approximated by piecewise constant functions. In the present situation we have developed new methods, which can be used in the framework of distributed or Neumann controls to consider piecewise linear approximations. This could lead to better error estimates than those deduced for piecewise controls.As far as we know, there is another paper dealing with the numerical approximation of a Dirichlet control problem of Navier-Stokes equations, by Gunzburger, *

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.

International audienceIn this paper we study optimal control problems governed by semilinear parabolic equations. We obtain necessary optimality conditions in the form of an exact Pontryagin's minimum principle for distributed and boundary controls (which can be unbounded) and bounded initial controls. These optimality conditions are obtained thanks to new regularity results for linear and nonlinear parabolic equations