Existence and characterization of an optimal control for the problem of long waves in a shallow-water model

In this paper we present a method of optimal control developed in order to calculate the current corresponding to the observed sea level in a fluid domain Omega and during a time T. The control is the external stress f. The cost function measures the distance between the observed and computed sea levels. The equations satis ed by the depth and the depth averaged velocity are of nonlinear shallow-water type. The existence and uniqueness of a solution for the direct problem are studied in the case of Dirichlet nonhomogeneous boundary conditions. We prove, by means of minimizing sequences, the existence of an optimal control (f, u) in the case of the small data and a very viscous fluid. To characterize it we build a sequence of problems corresponding to a linearization of the direct problem. We obtain the necessary conditions of optimality. The set of equations and the inequality characterizing the optimal control (f, u) is obtained as the limit of the penalization.