Optimal-Thrust Trajectories in an Arbitrary Gravitational Field

The problem of optimal-thrust trajectories is studied using a slight variation of the usual calculus of variations technique. The results include the usual first-order criteria for optimality, which are that the direction of thrust must be everywhere parallel to a solution ψ of the adjoint differential equation, and that the magnitude of the thrust must be zero in regions where the magnitude of ψ is less than a critical value, and equal to the maximum permissible value in regions where the magnitude of ψ is greater than the critical value. Singular arcs, on which the magnitude of ψ is continuously equal to the critical value, are shown to exist in the case of all except the simplest gravitational fields, and in some cases may form part of an optimal trajectory. A means of calculating the unique value of thrust required to sustain a singular arc is described, and a test for the optimality of such arcs is given. The test shows that a family of singular arcs discovered by D. F. Lawden is nonoptimal.