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

TL;DR

This paper establishes the existence of optimal controls for the 3D Primitive Equations of the ocean using Lagrangian observations, supported by new energy estimates and numerical validation.

Contribution

It introduces new energy estimates for the Primitive Equations and proves the existence of optimal controls in a regularized setting, with numerical illustration.

Findings

Existence of optimal control for the regularized Primitive Equations.

New energy estimates distinguishing vertical and horizontal dynamics.

Numerical experiment demonstrating theoretical results.

Abstract

We consider an optimal control problem for the three-dimensional non-linear Primitive Equations of the ocean in a vertically bounded and horizontally periodic domain. The observation operator maps a solution of the Primitive Equations to the trajectory of a Lagrangian particle. This paper proves the existence of an optimal control for the regularized problem. To do that, we prove also new energy estimates for the Primitive Equations, thanks to well-chosen functional spaces, which distinguish the vertical dimension from the horizontal ones. We illustrate the result with a numerical experiment.