Mathematical Programming: Series A and B
Global optimality in the calculus of variations
Nonlinear Analysis: Theory, Methods & Applications
Dynamic Programming
Some issues of search of extremal processes in nonconvex problems of optimal control
Automation and Remote Control
WSEAS Transactions on Systems and Control
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Let a trajectory and control pair (\bar{\mathbfx},\bar{\mathbf u}) maximize globally the functionalg(x(T)) in the basic optimal control problem. Then (evidently)any pair ({\mathbf x},{\mathbf u}) from the level set of thefunctional g corresponding to the valueg(\bar{x}(T)) is also globally optimal and satisfies thePontryagin maximum principle. It is shown that this necessary condition forglobal optimality of (\bar{\mathbf x},\bar{\mathbf u}) turnsout to be a sufficient one under the additional assumption of nondegeneracyof the maximum principle for every pair ({\mathbf x},{\mathbfu}) from the above-mentioned level set. In particular, if the pair(\bar{\mathbf x},\bar{\mathbf u}) satisfies the Pontryaginmaximum principle which is nondegenerate in the sense that for theHamiltonian H, we have along the pair (\bar{\mathbfx},\bar{\mathbf u}) \max_u H \not \equiv \min_u H on [0,T], and if there is no another pair ({\mathbf x},{\mathbfu}) such that g(x(T))=g(\bar{x}(T)), then(\bar{\mathbf x},\bar{\mathbf u}) is a global maximizer.