  In this work, a theoretical framework is developed to study the control
constrained distributed optimal control of a stationary double diffusion model
presented in [Burger, Mendez, Ruiz-Baier, SINUM (2019), 57:1318-1343]. For the
control problem, as the source term belongs to a weaker space, a new
solvability analysis of the governing equation is presented using Faedo-
Galerkin approximation techniques. Some new minimal regularity results for the
governing equation are established on two and three-dimensional bounded
Lipschitz domains and are of independent interest. Moreover, we show the
existence of an optimal control with quadratic type cost functional, study the
Frechet differentiability properties of the control-to-state map and establish
the first-order necessary optimality conditions corresponding to the optimal
control problem.
