Hamilton-Jacobi equations on heterogeneous networks or structures.

Summary This thesis deals with the study of optimal control problems on networks (i.e., sets of subregions connected by junctions), for which different dynamics and different instantaneous costs are allowed in each subregion of the network. As in the more classical cases, one would like to be able to characterize the value function of such a control problem through a Hamilton-Jacobi-Bellman equation. However, the geometrical singularities of the domain, as well as the discontinuities of the data, do not allow us to apply the classical theory of viscosity solutions. In the first part of this thesis we prove that the value functions of optimal control problems defined on 1-dimensional networks are characterized by such equations. In the second part the previous results are extended to the case of control problems defined on a 2-dimensional junction. Finally, in a last part, we use the previous results to treat a singular perturbation problem involving in-plane optimal control problems for which the dynamics and instantaneous costs can be discontinuous across an oscillating boundary.