Optimal control of a kinetic equation

TL;DR

This paper develops a stable finite element method for an optimal control problem constrained by a degenerate kinetic PDE, demonstrating well-posedness, hypocoercive decay, and validated effectiveness through numerical experiments.

Contribution

It introduces a hypocoercivity-based framework for stability analysis and a discretisation method that preserves these properties for kinetic PDE-constrained control problems.

Findings

Finite element discretisation preserves hypocoercive decay.

Method demonstrates stability and robustness in numerical tests.

Effective for challenging PDE-constrained optimal control problems.

Abstract

This work addresses an optimal control problem constrained by a degenerate kinetic equation of parabolic-hyperbolic type. Using a hypocoercivity framework we establish the well-posedness of the problem and demonstrate that the optimal solutions exhibit a hypocoercive decay property, ensuring stability and robustness. Building on this framework, we develop a finite element discretisation that preserves the stability properties of the continuous system. The effectiveness and accuracy of the proposed method are validated through a series of numerical experiments, showcasing its ability to handle challenging PDE-constrained optimal control problems.