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Abstract.In this paper, we study the stabilization of a two-dimensional Burgers equation around a stationary solution by a nonlinear feedback boundary control. We are interested in Dirichlet and Neumann boundary controls. In the literature, it has already been shown that a linear control law, determined by stabilizing the linearized equation, locally stabilizes the two-dimensional Burgers equation. In this paper, we define a nonlinear control law which also provides a local exponential stabilization of the two-dimensional Burgers equation. We end this paper with a few numerical simulations, comparing the performance of the nonlinear law with the linear one.Mathematics Subject Classification. 93B52, 93C20, 93D15.

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Nous nous intéressons à la stabilisation des équations de Navier-Stokes linéarisées autour d'une solution stationnaire instable, à l'aide d'un contrôle frontière dynamique. Nous utilisons une méthode de pénalisation afin d'approcher la condition d'incompressibilité. Nous obtenons un résultat de convergence de la solution des équations pénalisées vers la solution des équations incompressibles. Ces résultats sont ensuite validés par des tests numériques.

Let B be a n × n block diagonal matrix in which the first block C τ is an hermitian matrix of order (n − 1) and the second block c is a positive function. Both are piecewise smooth in Ω, a bounded domain of R n . If S denotes the set where discontinuities of C τ and c can occur, we suppose that Ω is stratified in a neighborhood of S in the sense that locally it takes the form Ω × (−δ, δ) with Ω ⊂ R n−1 , δ > 0 and S = Ω × {0}. We prove a Carleman estimate for the elliptic operator A = −∇ · (B∇ ) with an arbitrary observation region. This Carleman estimate is obtained through the introduction of a suitable mesh of the neighborhood of S and an associated approximation of c involving the Carleman large parameters.