  We consider the optimal control of solutions of first order Hamilton-Jacobi
equations, where the Hamiltonian is convex with linear growth. This models the
problem of steering the propagation of a front by constructing an obstacle. We
prove existence of minimizers to this optimization problem as in a relaxed
setting and characterize the minimizers as weak solutions to a mean field game
type system of coupled partial differential equations. Furthermore, we prove
existence and partial uniqueness of weak solutions to the PDE system. An
interpretation in terms of mean field games is also discussed.
  Keywords: Hamilton-Jacobi equations, optimal control, nonlinear PDE,
viscosity solutions, front propagation, mean field games
