Analysis of Unregularized Optimal Control Problems Constrained by the Boussinesq System

This paper investigates solution stability properties of unregularized tracking-type optimal control problems constrained by the Boussinesq system. In our model, the controls may appear linearly and distributed in both of the equations that constitute the Boussiniesq system and in the objective functional. We establish, not only the existence of weak solutions, but also unique existence of strong solutions in $L^p$ sense for the Boussinesq system as well as its corresponding linearized and adjoint systems. The optimal control problem is then analyzed by providing the existence of an optimal control, and by establishing first-order necessary and second order sufficient conditions. Then, using assumptions on the joint growth of the first and second variations of the objective functional, we prove the strong metric H\"older subregularity of the optimality mapping, which in turn allows the study of solution stability of the optimal control and states under various linear and nonlinear perturbations. Such perturbations may appear in the Boussinesq system and the objective functional. As an application, we provide a convergence rate for the optimal solutions of the Tikhonov regularized problem as the Tikhonov parameter tends to zero. Furthermore, the obtained stability of the optimal states provides, to the best knowledge of the authors, the first result on the stability of the second-order sufficient condition in affine PDE-constrained optimization under an assumption on the desired profile which is natural for tracking-type objective functionals.