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The long time behavior and detailed convergence analysis of Langevinequations has received increased attention over the last years. Difficultiesarise from a lack of coercivity, usually termed hypocoercivity, of theunderlying kinetic Fokker-Planck operator which is a consequence of thepartially deterministic nature of a second order stochastic differentialequation. In this manuscript, the effect of controlling the confinementpotential without altering the original invariant measure is investigated. Thisleads to an abstract bilinear control system with an unbounded butinfinite-time admissible control operator which, by means of an artificialdiffusion approach, is shown to possess a unique solution. The compactness ofthe underlying semigroup is further used to define an infinite-horizon optimalcontrol problem on an appropriately reduced state space. Under smallnessassumptions on the initial data, feasibility of and existence of a solution tothe optimal control problem are discussed. Numerical results based on a localapproximation based on a shifted Riccati equation illustrate the theoreticalfindings.

Improving the Convergence Rates for the Kinetic Fokker-Planck Equation by Optimal Control