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We consider an optimal control problem for the three-dimensional non-linearPrimitive Equations of the ocean in a vertically bounded and horizontallyperiodic domain. The observation operator maps a solution of the PrimitiveEquations to the trajectory of a Lagrangian particle. This paper proves theexistence of an optimal control for the regularized problem. To do that, weprove also new energy estimates for the Primitive Equations, thanks towell-chosen functional spaces, which distinguish the vertical dimension fromthe horizontal ones. We illustrate the result with a numerical experiment.

Optimal control of the Primitive Equations of the Ocean with Lagrangian observations