  The long time behavior and detailed convergence analysis of Langevin
equations has received increased attention over the last years. Difficulties
arise from a lack of coercivity, usually termed hypocoercivity, of the
underlying kinetic Fokker-Planck operator which is a consequence of the
partially deterministic nature of a second order stochastic differential
equation. In this manuscript, the effect of controlling the confinement
potential without altering the original invariant measure is investigated. This
leads to an abstract bilinear control system with an unbounded but
infinite-time admissible control operator which, by means of an artificial
diffusion approach, is shown to possess a unique solution. The compactness of
the underlying semigroup is further used to define an infinite-horizon optimal
control problem on an appropriately reduced state space. Under smallness
assumptions on the initial data, feasibility of and existence of a solution to
the optimal control problem are discussed. Numerical results based on a local
approximation based on a shifted Riccati equation illustrate the theoretical
findings.

Optimization and Control