Optimal boundary control of a system describing thermal convection

A problem of optimal boundary control of thermal sources for a stationary model of natural thermal convection of a high-viscosity inhomogeneous incompressible fluid in the Boussinesq approximation is investigated. Some conditions for the solvability of the problem are given; necessary and sufficient optimality conditions are specified. Optimality conditions and the corresponding conjugate problems defining the gradient of the quality functional are written for a number of particular cases of the functional. Procedures for the numerical finding of an optimal control based on gradient methods are described. The results of numerical experiments are given.