Singular characteristics in dynamic programming

The classical method of characteristics (MC) is a powerful tool for solving nonlinear first order PDEs arising in control theory and mathematical physics. The nonsmoothness of the generalized solution and/or of the Hamiltonian is often referred to as an obstacle for the implementation of the MC. However, one can overcome this obstacle using the same notion of characteristics suitably modified. In the paper a new notion of singular characteristics (SC) is introduced, which together with the classical (regular) characteristics allow one to construct a non-smooth solution for the Hamilton-Jacobi-Bellman-Isaacs equation. SC give an invariant description of the singular paths, surfaces and manifolds known in optimal control and differential games. The equations for singular characteristics in the form of closed ODE systems are derived for several types of singular surfaces. The method of SC is illustrated on a particular nonlinear PDE