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It is well known that Pontryagin’s maximum principle furnishes neces, sary conditions for the optimality of the control of a dynamic system. In the present work sufficient conditions for the optimality of the control of a nonlinear system with state and control variable constraints and with fixed initial and terminal times are given. These conditions are essentially Pontryagin’s necessary conditions for the same problem, plus some convexity, negativity and strict negativity conditions. The present sufficient conditions subsume the recent results of Lee, wherein sufficient conditions for the optimality of a system, linear in the state variables, were given.
O. L. Mangasarian (Tue,) studied this question.