Abstract

The transient and long time behavior 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. Similar challenges arise within the study of the continuous version of (stochastic) Cucker-Smale type flocking models where the resulting hypocoercive PDE additionally becomes nonlinear. Introducing controls on the finite-dimensional level naturally lead to abstract infinite-dimensional bilinear control problems with an unbounded but admissible control operator. By means of an artificial diffusion approach, solutions to a class of hypoercives PDEs as well as to associated optimal control problems are analyzed under smallness assumptions on the initial data.