  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.
