“Singular Extremals In Lawden’s Problem Of Optimal Rocket Flight”, 1963-07-01 ():
The problem of optimal rocket flight in an inverse square law force field has been studied extensively by Lawden and Leitmann. Periods of zero thrust, intermediate thrust, and maximum thrust are possible subarcs of the solution according to analysis of the Euler-Lagrange equations and the Weierstrass necessary condition. Arcs of intermediate thrust have been examined recently by Lawden; however, the question of whether or not such arcs actually may furnish a minimum has been left unresolved.
The present paper derives the singular extremals of Lawden’s problem by means of the Legendre-Clebsch necessary condition applied in a transformed system of state and control variables.
These are obtained as circular orbits along which the thrust is zero and intermediate thrust arcs are found in Lawden’s analysis. Since these solutions satisfy only the weak form of the Legendre-Clebsch condition, ie. the extremals are singular in the transformed system of variables, the question of their minimality remains unanswered.