Optimal Borel measure controls for the two-dimensional stationary Boussinesq system

We analyze an optimal control problem for the stationary two-dimensional Boussinesq system with controls taken in space of regular Borel measures. Such measure-valued are known to produce sparse solutions. First-order and second-order necessary sufficient optimality conditions established. Following optimize-then-discretize strategy, corresponding finite element approximation will be solved by a semi-smooth Newton method initialized continuation strategy. The discretized linear combinations Dirac measures concentrated at nodes associated degrees freedom mini-finite element.