Computational methods for optimal control and gradient flows in collective dynamics

This thesis develops and analyses numerical schemes for optimal control problems and their extensions, with a particular emphasis on preserving the analytical structure of the continuous models while ensuring computational efficiency. Optimal control theory, originating from the foundational works of Pontryagin and Bellman, offers powerful frameworks for regulating dynamical systems, yet its practical implementation faces significant challenges in the presence of non-linearity, high dimensionality, and non-convexity. Moreover, the recent emergence of collective control paradigms, such as mean field games, further complicates the numerical landscape by coupling individual optimisation with collective dynamics. The research presented here addresses these difficulties through four complementary contributions. First, a hybrid approach combining the global convergence of Dynamic Programming with the efficiency of the Pontryagin Maximum Principle is introduced for non-convex control problems, demonstrating improved robustness on epidemiological models. Second, a structure-preserving semi-Lagrangian scheme is proposed for systems governed by production–destruction dynamics, ensuring physical consistency and enabling high-performance implementations on GPUs. Third, variational image segmentation and colour quantisation are reformulated as optimal control problems for interacting particle systems, bridging ideas from opinion dynamics and image processing. Finally, a novel minimising movement formulation for first-order mean field games is developed, interpreting their dynamics as gradient flows in the space of probability measures endowed with the total variation distance. Through these contributions, the thesis advances the synthesis between variational analysis, optimal control, and scientific computing. The proposed schemes exemplify how discrete numerical methods can retain the geometric and energy-dissipative properties of continuous models, fostering a unified computational framework for modern control and optimisation problems across the sciences.