On the structure of singular set of a piecewise smooth minimax solution of Hamilton-Jacobi-Bellman equation

The properties of a minimax piecewise smooth solution of the Hamilton-Jacobi-Bellman equation are studied. We get a generalization of the nesessary and sufficient conditions for the points of nondifferentiability (singularity) of the minimax solution and the Rankine-Hugoniot condition. We describe the dimensions of smooth manifolds containing in the singular set of the piecewise smooth solution in terms of state characteristics crossing on this singular set. New structural properties of the singular set are obtained for the case of the Hamiltonian depending only on the impulse variable.