A first-order analytical theory for optimal low-thrust limited-power transfers between arbitrary elliptical coplanar orbits. (English) Zbl 1354.70053

Summary: A complete first-order analytical solution, which includes the short periodic terms, for the problem of optimal low-thrust limited-power transfers between arbitrary elliptic coplanar orbits in a Newtonian central gravity field is obtained through canonical transformation theory. The optimization problem is formulated as a Mayer problem of optimal control theory with Cartesian elements-position and velocity vectors-as state variables. After applying the Pontryagin maximum principle and determining the maximum Hamiltonian, classical orbital elements are introduced through a Mathieu transformation. The short periodic terms are then eliminated from the maximum Hamiltonian through an infinitesimal canonical transformation built through Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton-Jacobi equation through separation of variables technique. For transfers between close orbits a simplified solution is straightforwardly derived by linearizing the new Hamiltonian and the generating function obtained through Hori method.


70M20 Orbital mechanics
49K15 Optimality conditions for problems involving ordinary differential equations
49N90 Applications of optimal control and differential games
70F15 Celestial mechanics
70Q05 Control of mechanical systems
Full Text: DOI


[1] J.-P. Marec, Optimal Space Trajectories, Elsevier, New York, NY, USA, 1979. · Zbl 0435.70029
[2] G. Hori, “Theory of general perturbations with unspecified canonical variables,” Publications of the Astronomical Society of Japan, vol. 18, no. 4, pp. 287-296, 1966.
[3] C. Lánczos, The Variational Principles of Mechanics, Mathematical Expositions, no. 4, University of Toronto Press, Toronto, Canada, 4th edition, 1970. · Zbl 0257.70001
[4] T. N. Edelbaum, “Optimum power-limited orbit transfer in strong gravity fields,” AIAA Journal, vol. 3, pp. 921-925, 1965.
[5] J.-P. Marec and N. X. Vinh, “Optimal low-thrust, limited power transfers between arbitrary elliptical orbits,” Acta Astronautica, vol. 4, no. 5-6, pp. 511-540, 1977.
[6] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, NY, USA, 1962. · Zbl 0102.32001
[7] S. Da Silva Fernandes, “Generalized canonical systems-III. Space dynamics applications: solution of the coast-arc problem,” Acta Astronautica, vol. 32, no. 5, pp. 347-354, 1994.
[8] R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Education Series, American Institute of Aeronautics and Astronautics, Washington, DC, USA, 1987. · Zbl 0892.00015
[9] T. N. Edelbaum, “An asymptotic solution for optimum power limited orbit transfer,” AIAA Journal, vol. 4, no. 8, pp. 1491-1494, 1966. · Zbl 0212.57004
[10] G. E. Forsythe, M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1977. · Zbl 0361.65002
[11] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, vol. 12 of Texts in Applied Mathematics, Springer, New York, NY, USA, 3rd edition, 2002. · Zbl 1004.65001
[12] G. A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill, USA, 1946. · Zbl 0063.00459
[13] J.-P. Marec and N. X. Vinh, Étude Générale des Transferts Optimaux à Poussée Faible et Puissance Limitée entre Orbites Elliptiques Quelconques, ONERA, Châtillon, France, 1980.
[14] C. M. Haissig, K. D. Mease, and N. X. Vinh, “Minimum-fuel power-limited transfers between coplanar elliptical orbits,” Acta Astronautica, vol. 29, no. 1, pp. 1-15, 1993.
[15] T. N. Edelbaum, “Optimum low-thrust rendezvous and station keeping,” AIAA Journal, vol. 2, no. 7, pp. 1196-1201, 1964. · Zbl 0124.39704
[16] J.-P. Marec, Transferts Optimaux Entre Orbites Elliptiques Proches, ONERA, Châtillon, France, 1967.
[17] S. Da Silva Fernandes, “Optimum low-thrust limited power transfers between neighbouring elliptic non-equatorial orbits in a non-central gravity field,” Acta Astronautica, vol. 35, no. 12, pp. 763-770, 1995.
[18] S. Da Silva Fernandes, “Notes on Hori method for canonical systems,” Celestial Mechanics & Dynamical Astronomy, vol. 85, no. 1, pp. 67-77, 2003. · Zbl 1062.70037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.