Discontinuous Dynamical Systems in Optimal Control Theory

We develop foundations of the theory of discontinuous Hamiltonian systems appearing in the problems of optimal control. We consider analogs of the classical Poisson and Liouville theorems for discontinuous Hamiltonian systems. We study the local geometry of discontinuous dynamical systems and describe singularities in general position and the behavior of integral trajectories near an elliptical submanifold (sliding mode).