SOME LOCAL AND NONLOCAL OPTIMAL CONTROL PROBLEMS

This thesis presents three key contributions to optimal control theory, addressing problems with state constraints, mean field games, and Hamiltonian regularity. Chapter 1 tackles the problem of ensuring value function continuity for control systems with state constraints. When the standard Inward Pointing Condition (IPC) fails, discontinuities can arise. This work introduces a novel method using Lie brackets and a refined "rotational control" strategy with a time-varying, unbounded angular velocity. A significant improvement is the ability to handle systems with an uncontrolled drift. By establishing new second-order inward pointing conditions, the thesis proves the existence of "neighboring feasible trajectories"—constrained trajectories that remain close to an unconstrained reference. This directly leads to the continuity of the value function up to the constraint boundary, generalizing previous driftless results. Chapter 2 investigates the turnpike property for nonlocal mean field optimal control problems. This property describes how optimal trajectories of dynamic systems over long horizons spend most of their time near a steady state. The thesis establishes an exponential turnpike theorem for both Lagrangian (agent trajectory) and Eulerian (probability distribution) formulations. By deriving necessary optimality conditions and providing transformations between the two frameworks, it is shown that the turnpike property preserving under the transformation. Chapter 3 analyzes the regularity of Hamiltonian selectors, which are essential for the stability of mean field game equations. The research provides simple, verifiable criteria for these selectors to be Lipschitz or Hölder continuous. Under assumptions of uniform convexity, the work bridges convex analysis with practical control implementation, ensuring the robustness of optimal strategies in mean field models.