Necessary Conditions Joining Optimal Singular and Nonsingular Subarcs

Abstract

Necessary conditions for the optimality of junctions between singular and nonsingular subarcs are developed for singular optimal control problems. Previously known necessary conditions concerning the continuity and smoothness of a piecewise analytic optimal control at a junction are clarified and extended. The main result is that the sum of the order of the singular arc and the lowest order time derivative of the control which is discontinuous at the junction must be an odd integer when the strengthened generalized Legendre-Clebsch condition is satisfied. Also, new necessary conditions which do not require an analyticity assumption are developed. These aid in characterizing problems which may possess nonanalytic junctions.