AN OPTIMAL CONTROL PROBLEM FOR THE STATIONARY NAVIER–STOKES EQUATIONS WITH POINT SOURCES

Abstract. The aim of this work is to analyze a two dimensional 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 within the theory 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.

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