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

Analysis of PDEs