The author considers the following control problem in \({\mathbb{R}}^ n:\) minimize g(x(T)) over the solutions to the control system \(x'(t)=f(x(t),u(t))\) a.e. in [0,T], u(t)\(\in U\) is measurable selection, x(0)\(\in C.\) Let R(t,C) denote its reachable set at time t from the set of initial conditions \(C\subset {\mathbb{R}}^ n\) and \(T_{R(t,C)}(x_ 0)\) the contingent cone to R(t,C) at \(x_ 0\in {\mathbb{R}}^ n\). She studies some high-order necessary conditions for optimality via properties of contingent cones to reachable sets along the optimal trajectory and shows that the adjoint vector of Pontryagin's maximum principle is normal to the set of variations of the 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. The examples of optimal control problems in \({\mathbb{R}}^ 2\) and differential inclusion are considered.