  We introduce a novel Lyapunov function for stabilization of linear
Vlasov--Fokker--Planck type equations with stiff source term. Contrary to
existing results relying on transport properties to obtain stabilization, we
present results based on hypocoercivity analysis for the Fokker--Planck
operator. The existing estimates are extended to derive suitable feedback
boundary control to guarantee the exponential stabilization. Further, we study
the associated macroscopic limit and derive conditions on the feedback boundary
control such that in the formal limit no boundary layer exists.
