On global optimality conditions for nonlinear optimal control problems

Let a trajectory and control pair ((x) over bar, (u) over bar) maximize globally the functional g(x(T)) in the basic optimal control problem. Then (evidently) any pair (x, u) from the level set of the functional g corresponding to the value g((x) over bar(T)) is also globally optimal and satisfies the Pontryagin maximum principle. It is shown that this necessary condition for global optimality of ((x) over bar, (u) over bar) turns out to be a sufficient one under the additional assumption of nondegeneracy of the maximum principle for every pair (x, u) from the above-mentioned level set. In particular, if the pair ((x) over bar, (u) over bar) satisfies the Pontryagin maximum principle which is nondegenerate in the sense that for the Hamiltonian H, we have along the pair ((x) over bar, (u) over bar) [GRAPHICS] and if there is no another pair (x, u) such that g(x(T)) = g((x) over bar(T)), then ((x) over bar, (u) over bar) is a global maximizer.