Optimal control for the stationary Navier-Stokes equation involving the pressure, stress and their pointwise evaluations

We consider optimal control problems for the two-dimensional stationary Navier–Stokes equation with cost functionals involving the pressure, stress, diffusion, and convection. Likewise, we study observations for the velocity, pressure, and stress, concentrated on a finite collection of points located either in the domain or on the boundary. Such observations are known to produce PDEs with measure data for the corresponding adjoint equation. Depending on the nature of the objective cost functional, the control set will be either the space of square-integrable functions without constraints, with constraints in Lebesgue spaces, or with weak derivatives. We prove generalized Green's theorems for the weak and very weak solutions that involve the trace and normal stress on the boundary. Finally, the local optimality systems for such control problems will be derived and the regularity of the optimal states will be established.