In this thesis, the well-posedness of partial differential equations and optimal control problems are studied. The Cauchy problems associated with hyperbolic conservation laws with nonlocal velocities are studied first for a 1D model (manufacturing system) and then for a 2D model (process of follicular selection). In both cases, the existence and uniqueness of the solutions to the Cauchy problems are proved by Banach fixed point theorem. Optimal control problems on the 2D model and on an ODE-based model (amplification of misfolded proteins) are then studied. In the first model, optimal controls are shown to be bang-bang with one single switching time. In the second model, the optimal controls are relaxed controls which are localized on the admissible domain.
