Acta Applicandae Mathematicae

, Volume 135, Issue 1, pp 47–80 | Cite as

Time Versus Energy in the Averaged Optimal Coplanar Kepler Transfer Towards Circular Orbits

  • Bernard Bonnard
  • Helen C. Henninger
  • Jana Němcová
  • Jean-Baptiste PometEmail author


This article makes a study of the averaged optimal coplanar transfer towards circular orbits. Our objective is to compare this problem when the cost minimized is transfer time to the same problem when the cost minimized is energy consumption. While the minimum energy case leads to the analysis of a 2D-Riemannian metric using the standard tools of Riemannian geometry, the minimum time case is associated with a Finsler metric which is not smooth. Nevertheless a qualitative analysis of the geodesic flow is given in this article to describe the optimal transfers of the time minimal case.


Averaging Optimal control Low thrust orbit transfer Geodesic convexity Riemann-Finsler geometry 


  1. 1.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Graduate Texts in Mathematics, vol. 60. Springer, New York (1989). Translated from Russian by K. Vogtmann and A. Weinstein CrossRefGoogle Scholar
  2. 2.
    Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics, vol. 200. Springer, New York (2000) zbMATHGoogle Scholar
  3. 3.
    Bombrun, A., Pomet, J.B.: The averaged control system of fast oscillating control systems. SIAM J. Control Optim. 51(3), 2280–2305 (2013). doi: 10.1137/11085791X. CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bonnard, B., Caillau, J.B.: Geodesic flow of the averaged controlled Kepler equation. Forum Math. 21(5), 797–814 (2009). doi: 10.1515/FORUM.2009.038 CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bonnard, B., Sugny, D.: Optimal Control with Applications in Space and Quantum Dynamics. AIMS Series on Applied Mathematics, vol. 5. AIMS, Springfield (2012) zbMATHGoogle Scholar
  6. 6.
    Bonnard, B., Caillau, J.B., Dujol, R.: Energy minimization of single input orbit transfer by averaging and continuation. Bull. Sci. Math. 130(8), 707–719 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bonnard, B., Faubourg, L., Trélat, E.: Mécanique Céleste et Contrôle des Véhicules Spatiaux. Mathématiques & Applications, vol. 51. Springer, Berlin (2006) zbMATHGoogle Scholar
  8. 8.
    Edelbaum, T.N.: Optimum low-thrust rendezvous and station keeping. AIAA J. 2, 1196–1201 (1964) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Edelbaum, T.N.: Optimum power-limited orbit transfer in strong gravity fields. AIAA J. 3, 921–925 (1965) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Mathematics and Its Applications (Soviet Series), vol. 18. Kluwer Academic, Dordrecht (1988). doi: 10.1007/978-94-015-7793-9. Translated from the Russian Google Scholar
  11. 11.
    Geffroy, S.: Généralisation des techniques de moyennation en contrôle optimal – Application aux problèmes de rendez-vous orbitaux en poussée faible. Thèse de doctorat, Institut National Polytechnique de Toulouse, Toulouse, France (1997) Google Scholar
  12. 12.
    Geffroy, S., Epenoy, R.: Optimal low-thrust transfers with constraints—generalization of averaging techniques. Acta Astronaut. 41(3), 133–149 (1997). doi: 10.1016/S0094-5765(97)00208-7 CrossRefGoogle Scholar
  13. 13.
    Hartman, P.: Ordinary Differential Equations, 2nd edn. Birkhäuser, Basel (1982) zbMATHGoogle Scholar
  14. 14.
    Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, San Diego (2004) zbMATHGoogle Scholar
  15. 15.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: In: Neustadt, L.W. (ed.) The Mathematical Theory of Optimal Processes. Interscience/Wiley, New York/London (1962). Translated from the Russian by K.N. Trirogoff Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Bernard Bonnard
    • 1
  • Helen C. Henninger
    • 2
  • Jana Němcová
    • 3
  • Jean-Baptiste Pomet
    • 2
    Email author
  1. 1.Institut de Mathématiques de BourgogneUniversité de BourgogneDijonFrance
  2. 2.Team McTAOINRIA Sophia Antipolis MéditerranéeSophia Antipolis cedexFrance
  3. 3.Department of MathematicsInstitute of Chemical TechnologyPrague 6Czech Republic

Personalised recommendations