## Geometric interpretation of adjoint equations in optimal low thrust trajectories

S. Pifko, A. Zorn, and M. West

in *Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit*, 2008.

Time-optimal control of two seemingly unrelated
problems are solved using Pontryagin's Maximum Principle. The
first is a simple double integrator in ℜ^{2} in
which the state is driven to a desired terminal state in minimum
time. The second is an orbiting spacecraft in ℜ^{2}
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.

DOI: 10.2514/6.2008-6954

Full text: PiZoWe2008.pdf