NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMALITY

For the last few years, an enormous number of research publications on periodic processes have appared in the literature (see Fjeld [1], Guardabassi, Locatelli and Rinaldi [2] for the publications up to 1971). As the theory has reached a certain degree of growth and perfection, certain necessary and sufficient conditions have become available. Because of the peculiar behaviour of periodic processes, however, any extension of necessary and sufficient conditions of optimality for non-periodic processes to periodic processes is neither trivial nor straight-forward. The main difficulty is in the matching of the periodicity boundary conditions. This brings us to the problem of necessary and sufficient conditions for the existence and uniqueness of periodic solutions, a problem which was not severe in non-periodic cases. With this in mind, certain necessary and sufficient conditions for the existence and uniqueness of periodic solutions are first presented. Then the necessary conditions for optimality and the sufficient conditions, in particular the Hamilton-Jacobi theory, are covered. Certain conditions of relaxed controls are also briefly treated in the last section.

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