Speaker
Description
We consider a class of finite horizon deterministic mean field games with nonlocal coupling where the agents must follow Grushin type dynamics with state constraints. We require some assumptions on the local interplay between the set of state constraints and the dynamics.
As a first step, we consider the associated optimal control problem and we establish some properties as: the existence of an optimal trajectory for any starting point $(x,t)$, the closed graph property for the multivalued map which associates to each point $(x,t)$ the set of optimal trajectories starting from that point, the continuity of the value function.
Moreover, using a penalization method and the Maximum Principle, we obtain an uniform bound for the optimal controls. This allows us to obtain a comparison principle and that the value function is the unique constrained viscosity solution of the associated HJ equation.
Afterwards, we tackle the mean field games; taking advantage of the aforementioned properties, we prove the existence of a relaxed equilibrium (which describes the evolution of the game in terms of a probability on the set of admissible trajectories) and derive the existence of a mild solution (which is a couple formed by the value function for the generic player and a family of time dependent measures on the state).
Research project in collaboration with Alessandra Cutri' (Roma Tor Vergata), Claudio Marchi (Universita' di Padova), Nicoletta Tchou (Universite' de Rennes)