Junction point on partially singular trajectories

Abstract

In recent times, several works have been performed on the design of fuel optimal trajectories for space navigation. These works show the possibility of the existence of partially singular trajectories for systems that are linear analytic (Park et al. 2010). Linear analytic systems may show the existence of partially singular subarcs, and the point where these subarcs meet is called a junction point. Thus, knowledge about junction conditions became necessary when solving the optimal control problem for such systems. This led to the development of two ‘theorems’ on junction conditions, given by McDanell and Powers (McDanell, J.P. and Powers W.F. (1971), ‘Necessary Conditions for Joining Optimal Singular and Nonsingular Sub Arcs’, SIAM Journal of Control, 9, 161–173). However, the second ‘theorem’, which is now known as a conjecture, could not satisfy all classes of linear analytic system. Therefore, the aim of this study was to detect and correct the errors in the derivation of the McDanell and Powers conjecture. The error in their derivations was corrected and then tested on two newly mathematically constructed systems. The results of these tests were found to be satisfactory. This implies that by making the necessary corrections, the conjecture can still be useful in generating a general theorem for all classes of systems.