Terminality,normality, and transversality conditions

The maximum principle distinguishes between two phases of the optimal control problem. Some of the stated conditions are to be satisfied at points other than endpoints and some conditions are to be satisfied specifically at the endpoints. This paper utilizes the first set of conditions from the maximum principle to reexamine the second set. In the process, a new necessary condition to be satisfied at the endpoints is obtained. This condition is for certain cases easier to apply than the transversality conditions, yields additional information which may be of computational advantage, and lends itself quite naturally to an exposition of abnormal solutions. The relationship of the new condition to the transversality conditions and a discussion on normality are included. Several examples are given to illustrate the results.This research was supported in part by NASA under Grant NGR-03-002-224 and NSF Science Faculty Fellowship. The authors are indebted to Professors G. Leitmann, G. Basile, and E. Cliff for their helpful comments and suggestions.