  This paper is concerned with a bilinear control problem for enhancing
convection-cooling via an incompressible velocity field. Both optimal open-loop
control and closed-loop feedback control designs are addressed. First and
second order optimality conditions for characterizing the optimal solution are
discussed. In particular, the method of instantaneous control is applied to
establish the feedback laws. Moreover, the construction of feedback laws is
also investigated by directly utilizing the optimality system with appropriate
numerical discretization schemes. Computationally, it is much easier to
implement the closed-loop feedback control than the optimal open-loop control,
as the latter requires to solve the state equations forward in time, coupled
with the adjoint equations backward in time together with a nonlinear
optimality condition. Rigorous analysis and numerical experiments are presented
to demonstrate our ideas and validate the efficacy of the control designs.
