Junction points in singular optimal control

Optimal control trajectories which contain both non-singular and singular subarcs have been known for almost 40 years. Points where these two types of subarc are joined are known as junction points. Necessary and sufficient conditions for the optimality of these so-called partially singular trajectories are well-known. Necessary conditions for optimality at junction points have also been found. However, for many years the status of McDanell's Conjecture has been unknown. One part of the Conjecture was proved false in 1987 but the remaining part has been hard to prove or disprove. There has been much evidence that the whole of the Conjecture is false in general, although it is true in several special cases. This paper presents a counterexample for the remaining part of McDanell's Conjecture. A minimum of nine equations and two inequalities had to be satisfied at the junction in order for the counterexample to be constructed. The resulting system is of third order and the problem is of Lagrange type.