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Time versus energy in the averaged optimal coplanar Kepler transfer towards circular orbits. (English) Zbl 1309.70029
Summary: 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.
##### MSC:
 70Q05 Control of mechanical systems 70F15 Celestial mechanics 93C70 Time-scale analysis and singular perturbations in control/observation systems 53C80 Applications of global differential geometry to the sciences
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##### References:
 [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 [2] Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics, vol. 200. Springer, New York (2000) · Zbl 0954.53001 [3] Bombrun, A.; Pomet, J.B., The averaged control system of fast oscillating control systems, SIAM J. Control Optim., 51, 2280-2305, (2013) · Zbl 1282.34067 [4] Bonnard, B.; Caillau, J.B., Geodesic flow of the averaged controlled Kepler equation, Forum Math., 21, 797-814, (2009) · Zbl 1171.49030 [5] Bonnard, B., Sugny, D.: Optimal Control with Applications in Space and Quantum Dynamics. AIMS Series on Applied Mathematics, vol. 5. AIMS, Springfield (2012) · Zbl 1266.49002 [6] Bonnard, B.; Caillau, J.B.; Dujol, R., Energy minimization of single input orbit transfer by averaging and continuation, Bull. Sci. Math., 130, 707-719, (2006) · Zbl 1136.93046 [7] Bonnard, B., Faubourg, L., Trélat, E.: Mécanique Céleste et Contrôle des Véhicules Spatiaux. Mathématiques & Applications, vol. 51. Springer, Berlin (2006) · Zbl 1104.70001 [8] Edelbaum, T.N., Optimum low-thrust rendezvous and station keeping, AIAA J., 2, 1196-1201, (1964) · Zbl 0124.39704 [9] Edelbaum, T.N., Optimum power-limited orbit transfer in strong gravity fields, AIAA J., 3, 921-925, (1965) [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 [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) [12] Geffroy, S.; Epenoy, R., Optimal low-thrust transfers with constraints—generalization of averaging techniques, Acta Astronaut., 41, 133-149, (1997) [13] Hartman, P.: Ordinary Differential Equations, 2nd edn. Birkhäuser, Basel (1982) · Zbl 0476.34002 [14] Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, San Diego (2004) · Zbl 1135.37002 [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 · Zbl 0102.32001
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