Mean field optimal control with piecewise deterministic Markov processes

This work studies the theoretical resolution and numerical approximation of a mean field optimization problem. More precisely, a mean field optimal control problem of piecewise deterministic Markov processes is formulated, modeling the optimal charging of a large fleet of electric vehicles. Optimality conditions are obtained through a linearization procedure and are studied via a system of coupled partial differential equations, similar to those encountered in mean field games. The problem is numerically solved using the General Frank-Wolfe algorithm, a variant of the conditional gradient algorithm. This algorithm is analyzed, and it is proved that it has linear convergence and satisfies the mesh-independence property: its rate and the underlying convergence constants are independent of the discretization parameters.