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In this paper, the boundary flex control problem of non stationary equationgoverning the coupled mass and heat flow of a viscous incompressible fluid in ageneralized Boussinesq approximation by assuming that viscosity and heatconductivity are dependent on temperature has been studied. The boundarycondition for velocity of fluid is non -standard boundary condition:specifically the case where dynamical pressure is given on some part of theboundary and the boundary condition for temperature of fluid is mixed boundarycondition has been considered. The optimal condition has been derived. Then,Pontryagin's maximum principle in the special case has been derived.

Optimal condition of boundary flex control for the systems governed by  Boussinesq equation with the press boundary condition and mixed boundary  condition