A pointwise tracking optimal control problem for the stationary Navier-Stokes equations

We study a pointwise tracking optimal control problem for the stationary Navier-Stokes equations; control constraints are also considered. The control problem involves the minimization of a cost functional involving point evaluations of the state velocity field, which leads to an adjoint problem with a linear combination of Dirac measures as a forcing term in the momentum equation, and whose solution has reduced regularity properties. We analyze the existence of optimal solutions and derive first and, necessary and sufficient, second-order optimality conditions in the framework of regular solutions for the Navier-Stokes equations. We develop two discretization strategies: a semidiscrete strategy in which the control variable is not discretized, and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For each solution technique, we analyze convergence properties of discretizations and derive error bounds. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.