Hypocoercivity in Optimal Control with Kinetic Constraints

The application of hypocoercivity to optimal control problems with hyperbolic/parabolic kinetic equations as constraints. The aim of this analysis is to demonstrate exponential convergence of both analytical and numerical solutions to the dynamic equation to the steady state. Additionally, constructing uniform time dependent bounds on the norm of the dynamic solution.

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.

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