Contingent Cones to Reachable Sets of Control Systems

High-order necessary conditions for optimality for an optimal control problem are studied via properties of contingent cones to reachable sets along the optimal trajectory. It is shown that the adjoint vector of Pontryagin’s maximum principle is normal to the set of variations of reachable sets. Results are applied to study optimal control problems for dynamical systems described by: (1) closed-loop control systems; (2) nonlinear implicit systems; (3) differential inclusions; (4) control systems with jumps