Pointwise control of the Boussinesq system

We study a pointwise control problem for the Boussinesq system in two dimensions. The control is a source term in the heat equation, and the cost functional takes into account the distance between the solution to the Boussinesq system and a given profile. We have recently studied similar problems for the linearized Boussinesq system (Nguyen and Raymond, J. Optim. Theory Appl. 141 (2009) 147–165). Studying these problems in the nonlinear case is more difficult. There is some literature on the pointwise control of linear and nonlinear partial differential equations; however, to the authors’ knowledge, nothing has been done for the Boussinesq system. In this study, we prove the existence of optimal solutions and derive optimality conditions.