Optimal Control of Heat Transfer in Unsteady Stokes Flows

This paper is concerned with establishing a rigorous mathematical framework to address optimal control designs for heat transfer in unsteady Stokes flows. In particular, we focus on the problem of enhancing convection-cooling between two fluids via controlling the velocity of the cold fluid flow, where both internal and boundary controls will be investigated. This essentially leads to a bilinear control problem. We present a rigorous proof of the existence of an optimal control and derive the first-order necessary conditions for optimality by using a variational inequality. Finally, we show that the uniqueness of the optimal controller can be obtained if the control weight is sufficiently large.