Sufficient conditions in free final time optimal control problems. A comment (Q790417)

A few types of sufficient conditions for free final time optimal control problems have been stated, see \textit{P. M. Merean} and \textit{W. F. Powers} [SIAM J. Control Optimization 14, 613-622 (1976; Zbl 0331.49019)], \textit{A. J. Robson} [J. Econ. Theory 24, 437-445 (1981; Zbl 0469.49017)]. This note shows that the conditions of the latter paper are incorrect and proposes a new set of sufficient conditions, the content of which is essentially the following: If there exist optimal pairs in all fixed final time problems satisfying fixed final time sufficient conditions and if the corresponding Hamiltonians, with the final times inserted, equals zero for some fixed final time \(\tau^*\), while being \(\geq 0\) (\(\leq 0)\) to the left (right) of \(\tau^*\), then \(\tau^*\) and the pair corresponding to it, are optimal in the free final time problem. The proof of this note utilizes a uniqueness condition of the adjoint functions in all the fixed time problems. This condition, however, can be removed.