On Mcdanell's conjecture

Optimal control trajectories generated by optimal control problems in which the input appears linearly in the Hamiltonian may have singular subarcs. When optimal control trajectories include both nonsingular and singular subarcs they intersect at junction points. Optimality conditions at these junctions were first proposed by Mcdanell and Powers. However, it has been shown that the conditions do not hold for all problems, and therefore these conditions have later been referred to as Mcdanell's Conjecture. There have been numerous attempts to either prove or disprove this conjecture. In this paper, we summarise the findings related to Mcdanell's Conjecture, and clarify the ambiguous definitions of certain variables used in Mcdanell and Powers original work and subsequent literature. We also present a counter-example to a modified version of Mcdanell's Conjecture. Finally, we demonstrate that Mcdanell's Conjecture holds under an additional assumption.