On some control problems in a shallow-water model

We develop methods of control for the modelling of long waves occurring in shallow water zones. The observation is the variability of the sea level. It is deduced from satellite measurements. The equations satisfied by the depth and the depth averaged velocity are of nonlinear shallow-water type. The result of existence and uniqueness of the solution for the direct problem is given in the case of Dirichlet non-homogeneous boundary conditions. We establish, by means of minimizing sequences, the existence of an optimal control in the case of small data and a very viscous fluid. In order to characterize it, we build a sequence of control problems corresponding to a linearization of the problem. We obtain the necessary conditions of optimality. The optimality conditions for the nonlinear problem is obtained as the limit of the penalization. Our approach can be used to develop a numerical scheme