Hamilton-Jacobi equation and mean-field games on networks.

Summary This thesis deals with the study of Hamilton-Jacobi-Bellman equations associated with optimal control and mean-field games problems with the particularity that we place ourselves on a network (i.e., sets consisting of edges connected by junctions) in both problems, for which we allow different dynamics and different costs in each edge of a network. In the first part of this thesis, we consider an optimal control problem on networks in the spirit of the work of Achdou, Camilli, Cutrì & Tchou (2013) and Imbert, Moneau & Zidani (2013). The main novelty is that we add input (or output) costs to the vertices of the network leading to a possible discontinuity of the value function. This is characterized as the unique viscosity solution of a Hamilton-Jacobi equation for which a suitable join condition is established. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from optimal control theory, inspired by Achdou, Oudet & Tchou (2015) and the other based on partial differential equations, after Lions & Souganidis (2017). The second part concerns stochastic mean-field games on networks. In the ergodic case, they are described by a system coupling a Hamilton-Jacobi-Bellman equation and a Fokker- Planck equation, whose unknowns are the density m of the invariant measure that represents the distribution of players, the value function v that comes from an "average" optimal control problem, and the ergodic constant ρ. The value function v is continuous and satisfies very general Kirchhoff conditions at vertices in our problem. The function m satisfies two transmission conditions at vertices. In particular, due to the generality of Kirchhoff conditions, m is in general discontinuous at the vertices. The existence and uniqueness of a weak solution are proved for subquadratic Hamiltonians and very general assumptions on the coupling. Finally, in the last part, we study non-stationary stochastic mean field games on networks. The transition conditions for the value function v and the density m are similar to those given in the second part. Again, we prove the existence and uniqueness of a weak solution for sublinear Hamiltonians and couplings and in the case of a lower bounded regularizing nonlocal coupling. The main additional difficulty compared to the stationary case, which imposes more restrictive assumptions, is to establish the regularity of the solutions of the system on a lattice. Our approach consists in studying the solution of the Hamilton-Jacobi equation derived to gain regularity on the solution of the initial equation.