Robust control problems for the multilayer quasi-geostrophic system

In this article, we study some robust control problems associated with the multilayer quasi-geostrophic equations of the ocean and related to data assimilation in oceanography. We consider higher norms (compared to T. Tachim Medjo, Robust control problems associated with the multilayer quasi-geostrophic equations of the ocean, Appl. Math. Optim. 51(3) (2005) 333–360]) in the definition of the cost functionals. We prove the existence and uniqueness of solutions. The result relies on better a priori estimates on the solutions to the multilayer quasi-geostrophic system obtained using a new formulation that we introduce for the multilayer quasi-geostrophic equation of the ocean. The new formulation replaces the non-homogenous boundary conditions (and the non-local constraint) on the stream-function by a simple homogenous Dirichlet boundary condition.