Geometric interpretation of adjoint equations in optimal low thrust space flight

Steven L. Pifko, Alan H. Zorn, Matthew West

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Time-optimal control of two seemingly unrelated problems are solved using Pontryagin's Maximum Principle. The first is a simple double integrator in R2 in which the state is driven to a desired terminal state in minimum time. The second is an orbiting spacecraft in R2 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'sMaximum 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. American Institu

Original languageEnglish (US)
Title of host publicationAIAA/AAS Astrodynamics Specialist Conference and Exhibit
StatePublished - 2008
EventAIAA/AAS Astrodynamics Specialist Conference and Exhibit - Honolulu, HI, United States
Duration: Aug 18 2008Aug 21 2008

Publication series

NameAIAA/AAS Astrodynamics Specialist Conference and Exhibit


OtherAIAA/AAS Astrodynamics Specialist Conference and Exhibit
Country/TerritoryUnited States
CityHonolulu, HI

ASJC Scopus subject areas

  • Astronomy and Astrophysics


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