Abstract
The generalized Legendre–Clebsch higher order tests for optimality of singular arcs in optimal control problems depend upon the orders of the arcs involved. To date three distinct definitions of order have been given but many authors do not distinguish among them. The features of each definition are discussed with special reference to the applicability of the higher order tests and of the conditions at junctions between singular and nonsingular arcs; only in terms of one of the definitions are the junction conditions generally valid. An illustrative example is presented.
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