A nonlinear optimal control problem for assimilation of surface data in navier-stokestype equations related to oceanography

Abstract

This paper presents a method of control developed in order to calculate the currents corresponding to the pressure measured at the level of the undisturbed sea surface in an oceanic domain and during a time T. We use a perturbation model suitable fortropical oceans, where the mean flow is known and its variability is of great interest for oceanography. The equations satisfied by the variability of the velocity and pressure are of nonlinear Navier-Stokes type. The control is the variability of the wind-stress which acts as the forcing of the perturbation. The cost function measures the distance between the observed and computed pressures on a part of the boundary. We prove, by means of minimizing sequences, the existence of an optimal control . In order to characterize it, we define a penalized control problem corresponding to a linearization of the direct problem satisfied by . Then the optimal penalized control is characterized by a set of linear equations: direct and adjoint problems, linked by an inequality. The set of equations and inequality characterizing the optimal control is obtained as the limit of the penalization. The choice of the surface pressure as the observation makes the problem more difficult: it makes it necessary to use a mixed velocity-pressure formulation for the Navier-Stokes type problems and to satisfy regularity conditions in order to define the trace of the pressure on the boundary. Regularity must be satisfied by all intermediate problems and preserved for the limit.