An Optimal Control Problem for the Navier–Stokes Equations with Point Sources

Abstract

We analyze, in two dimensions, an optimal control problem for the Navier–Stokes equations where the control variable corresponds to the amplitude of forces modeled as point sources; control constraints are also considered. This particular setting leads to solutions to the state equation exhibiting reduced regularity properties. We operate under the framework of Muckenhoupt weights, Muckenhoupt-weighted Sobolev spaces, and the corresponding weighted norm inequalities and derive the existence of optimal solutions and first- and, necessary and sufficient, second-order optimality conditions.