Higher-order pointwise optimality conditions

Necessary higher-order optimality conditions are given for nonlinear smooth nonstationary control systems, linear with respect to control and defined on a smooth finite-dimensional manifold; the admissible values of the control belong to a closed convex polyhedron, while the initial and final times and the initial and final points of the trajectory are not fixed. Unlike the previously known necessary optimality conditions, the paper does not require that the passage from one condition, of a given sequence of necessary conditions, to another one should be performed only when the preceding condition degenerates on a time interval of nonzero length.