Existence and Characterization of an Optimal Control for the Problem of Long Waves in a Shallow-Water Model
Résumé
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 satisfied 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.