We study a pointwise tracking optimal control problem for the stationary Navier--Stokes equations; control constraints are also considered. The problem entails the minimization of a cost functional involving point evaluations of the state velocity field, thus leading 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 a priori error estimates.

翻译：本文研究驻定Navier-Stokes方程在控制约束下的定位置跟踪最优控制问题。该问题涉及含有状态速度场点评估的成本泛函最小化，从而在动量方程中引入Dirac测度线性组合作为强迫项的伴随问题，其解具有降低的正则性。我们在Navier-Stokes方程正则解框架下分析最优解的存在性，推导一阶及必要与充分二阶最优性条件。发展两种离散化策略：控制变量未被离散的半离散策略，以及控制变量采用逐片常数函数离散的全离散格式。针对每种求解技术，分析离散化的收敛性质并推导先验误差估计。