In dissipative systems, we may quantify the rate of convergence of the solution to the dynamic problem to the steady state; this is obtained by applying the Poincaré inequality. However, for semi-dissipative systems; such as the Boltzmann-Fokker Planck equations and the Kolmogorov equation, standard energy arguments involving testing the operator against the solution do not yield a bound with respect to the correct norm, and thus we do not obtain our time-dependent bound.

This serves as motivation for the hypocoercive formulation, as a means of yielding regularity of the gradient in directions where dissipation is not explicitly present by constructing a non-standard energy argument using mixed derivatives. The hypocoercive framework facilitates a stability bound more inline with the parabolic stability \eqref{eq:parabolicstability} through modifying the energy argument by assuming additional regularity on the source term and choosing specific test functions.

We introduce a hypocoercive framework to PDE constrained optimisation problems involving kinetic equations, with the goal being to demonstrate stability of both the analytical dynamical solution and numerical schemes with respect to time.