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.

翻译：本研究探讨了一个受退化抛物-双曲型动力学方程约束的最优控制问题。利用亚强制性框架，我们建立了问题的适定性，并证明了最优解具有亚强制性衰减特性，从而确保了系统的稳定性与鲁棒性。基于此框架，我们发展了一种有限元离散化方法，该方法保持了连续系统的稳定性特征。通过一系列数值实验，验证了所提方法的有效性与精确性，展示了其处理具有挑战性的偏微分方程约束最优控制问题的能力。