Optimal Control of Stationary Doubly Diffusive Flows on Lipschitz Domains

In this work, we study the control constrained distributed optimal control of a stationary doubly diffusive flow model. For the control problem, we use a well-posedness analysis based on minimal assumptions on data and domain. We show the existence of an optimal control with quadratic type cost functional, study the Fréchet differentiability properties of the control-to-state map and establish the first-order necessary optimality conditions corresponding to the optimal control problem. Expanding on this we prove the local optimality of a reference control using second-order sufficient optimality condition for the control problem.