Optimal boundary control in thermal fluid dynamics problems with different smoothness control

This paper deals with boundary optimal control for the energy equation coupled with Navier-Stokes system. We propose and compare three different approaches to solve the optimal control problem for this system of equations. The first approach is the standard approach resulting in a fully coupled optimality system with second order finite element approximation. In this case we do not compute directly the trace of the adjoint derivative but use a volumetric integral to enforce implicit Dirichlet conditions over the controlled boundary. In the second approach we search discontinuous solution by using standard Discontinuous Galerkin formulation for the energy transport equation. Also in this case we solve implicitly the fully coupled system. In the last approach we limit the controlled temperature to a more regular class of solutions with smooth derivatives and solve the resulting optimality system in a segregated way. We apply these three approaches to some test cases and compare the results obtained in term of smoothness of the solution and value of the objective functional