On singularities of value function for Bolza optimal control problem

We study the set of points of nondifferentiability, called the singular set, of the value function of a Bolza optimal control problem. The value function is a viscosity solution to an associated Hamilton-Jacobi equation. The method of characteristics associates to this equation a Hamiltonian system, that in turn can be used to study the propagation of singularities of the value function. In particular, we obtain an extension of the Rankine-Hugoniot type condition, which is well-known in the conservation law theory.