Geometric Interpretation of Adjoint Equations in Optimal Low Thrust Trajectories

Abstract

Time-optimal control of two seemingly unrelated problems are solved using Pontryagin’s Maximum Principle. The first is a simple double integrator in R in which the state is driven to a desired terminal state in minimum time. The second is an orbiting spacecraft in R which transitions from its current orbit into a desired terminal orbit in minimum time. In both cases, thrust is continuously available but limited in magnitude. The two problems are related by the gravitational parameter of the major body orbited. As the gravitational parameter is mathematically varied to zero, the orbiting spacecraft takes on the dynamics of a double integrator. A two-point boundary value problem is created when Pontryagin’s Maximum Principle is applied to solve the two problems. Shooting methods are typically used in the solution, but they require reasonably close a priori estimates of the initial or final values of the costate for the shooting method to converge. The adjoint equations of the double integrator have a simple solution. The derived optimal control is shown to be related to the adjoint solution in a simple geometric manner. A method is presented to estimate the initial costate and terminal time for the double integrator problem. The possibility that the initial estimate for the double integrator may provide an initial estimate for the related orbital transfer problem is explored. Numerical examples of the two problems illustrate the method.