A Control Method for Assimilation of Surface Data in a Linearized Navier--Stokes-Type Problem Related to Oceanography
Résumé
A method of control is developed in order to compute the variability of the velocity and pressure $(u, p)$ in an oceanic doman $\Omega$ during a time $T$. The observation is the variability of $p$ of $\Gamma_0 \times ]0, T[$, where $\Gamma_0$ is the upper surface of $\Omega$ and corresponds to the undisturbed sea surface. The observation is deduced from satellite data. The control is the variability of the wind-stress $f$, which acts as the forcing of the perturbation. The mean circulation in $\Omega$ is supposed to be known. The equations verified by $(u, p)$ are linearized around this mean circulation. The continuity of the application: $f \rightarrow (u(f), p(f))$ in convenient functional spaces is proved. We deduce from this result the existence and uniqueness of an optimal control, which is characterized by a set of equations including the direct problem of linearized Navier--Stokes type verified by $(u, p)$ and the adjoint problem. The cost function and consequently the optimal control depend on a real parameter $\alpha$. We prove the convergence of the sequence $(f_{\alpha})$, when $\alpha$ tends to 0, toward a wind-stress $f$ which minimizes the distance between the observed and computed pressures. This result is obtained by means of minimizing sequences.