Resolution numerique d'un probleme de controle optimal d'un couplage des equations de navier-stokes et celle de la chaleur

We study by the method of artificial compressibility the numerical solution of an optimal control problem of a system governed by the Navier-Stokes equations coupled with the heat-equation. The control problem is of boundary control—final observation type. A system of equations and an inequality characterizing the optimal control gives rise to an algorithm which permits us to calculate a control satisfying a necessary condition for optimality. The algorithm will be approximated by finite difference schemes based on the fractional step method. We prove the stability and the convergence of the schemes and the convergence of the algorithm. Finally we present some numerical results illustrating the proposed method.