  In this paper, we study the bilinear optimal stabilization of a
non-homogeneous Fokker-Planck equation. We first study the problem of optimal
control in the case of finite-time interval for non-homogeneous system, and
then focus on the case of infinite-time horizon. The main difficulty in solving
a quadratic optimal control for bilinear systems is the non-convexity of the
cost function, and the unboundedness of the operator control. We further show
that the obtained optimal control guarantees the strong stability of the
non-homogeneous unbounded system. The techniques we use rely on the theory and
conditions of optimality.
