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Shooting and numerical continuation methods for computing time-minimal and energy-minimal trajectories in the Earth-Moon system using low propulsion
1. | Mathematics Institute, Bourgogne University, 9 avenue Savary, 21078 Dijon, France |
References:
[1] |
A. A. Agrachev and A. V. Sarychev, On abnormals extremals for Lagrange variational problems, J. Math. Systems Estim. Control, 8 (1998), 87-118. |
[2] |
E. L Allgower and K. Georg, "Numerical Continuation Methods. An Introduction," Springer Series in Computational Mathematics, 13, Springer-Verlag, Berlin, 1990. |
[3] |
V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," Second edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. |
[4] |
J. T. Betts and S. O. Erb, Optimal low thrust trajectories to the moon, SIAM J. Appl. Dyn. Syst., 2 (2003), 144-170.
doi: 10.1137/S1111111102409080. |
[5] |
C. Bischof, A. Carle, P. Kladem and A. Mauer, Adifor 2.0: Automatic Differentiation of Fortran 77 programs, IEEE Computational Science and Engineering, 3 (1996), 18-32.
doi: 10.1109/99.537089. |
[6] |
G. A. Bliss, "Lectures on the Calculus of Variations," University of Chicago Press, Chicago, Ill., 1946. |
[7] |
A. Bombrun, J. Chetboun and J.-B. Pomet, Transfert Terre-Lune en poussée faible par contrôle feedback - La mission SMART-1, (French) INRIA Research Report, 5955 (2006), 1-27. |
[8] |
B. Bonnard, J.-B. Caillau and G. Picot, Geometric and numerical techniques in optimal control of the two and three-body problems, Commun. Inf. Syst., 10 (2010), 239-278. |
[9] |
B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM Control Optim. and Calc. Var., 13 (2007), 207-236. |
[10] |
B. Bonnard, J.-B. Caillau and E. Trélat, COTCOT: Short reference manual, ENSEEIHT-IRIT Technical Report RT/APO/05/1, (2005), 1-15. |
[11] |
B. Bonnard and M. Chyba, "Singular Trajectories and Their Role in Control Theory," Math. and Applications, 40, Springer-Verlag, Berlin, 2003. |
[12] |
B. Bonnard, L. Faubourg and E. Trélat, "Mécanique Céleste et Contôle des Véhicules Spatiaux," Mathématiques & Applications (Berlin), 51, Springer-Verlag, Berlin, 2006. |
[13] |
B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, (French)[Theory of the singularities of the input/output mapping and optimality of singular trajectories in the minimal-time problem], Forum Math., 5 (1993), 111-159.
doi: 10.1515/form.1993.5.111. |
[14] |
B. Bonnard, N. Shcherbakova and D. Sugny, The smooth continuation method in optimal control with an application to quantum systems, ESAIM Control Optim. and Calc. Var., 17 (2011), 267-292.
doi: 10.1051/cocv/2010004. |
[15] |
J.-B. Caillau, "Contribution à l'Etude du Contrôle en Temps Minimal des Transferts Orbitaux," Ph.D thesis, Toulouse University, 2000. |
[16] |
J.-B. Caillau, B. Daoud and J. Gergaud, On some Riemannian aspects of two and three-body controlled problems, in "Recent Advances in Optimization and its Applications in Engineering," Springer, (2010), 205-224. |
[17] |
J.-B. Caillau, B. Daoud and J. Gergaud, Discrete and differential homotopy in circular restricted three-body control, to appear in AIMS Proceedings, 2010. |
[18] |
J. Gergaud and T. Haberkorn, Homotopy method for minimum consumption orbit transfer problem, ESAIM Control Optim. Calc. Var., 12 (2006), 294-310.
doi: 10.1051/cocv:2006003. |
[19] |
G. Gómez, S. Koon, M. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, Invariant manifolds, the spatial three-body problem ans space mission design, in "The Proceedings of AIAA/AAS Astrodynamics Specialist Meeting," Quebec City, Quebec, Canada, 2001. |
[20] |
M. Guerra and A. Sarychev, Existence and Lipschitzian regularity for relaxed minimizers, in "Mathematical Control Theory and Finance," Springer, Berlin, (2008), 231-250. |
[21] |
J. E. Marsden and S. D. Ross, New methods in celestial mechanics and mission design, Bull. Amer. Math. Soc. (N.S), 43 (2006), 43-73. |
[22] |
K. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem," Applied Mathematical Sciences, 90, Springer-Verlag, New York, 1992. |
[23] | |
[24] |
H. Pollard, "Mathematical Introduction to Celestial Mechanics," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. |
[25] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962. |
[26] |
G. Racca, B. H. Foing and M. Coradini, SMART-1: The first time of Europe to the moon, Earth, Moon and Planets, 85-86 (2001), 379-390. |
[27] |
G. Racca, et al., SMART-1 mission description and development status, Planetary and Space Science, 50 (2002), 1323-1337. |
[28] |
A. V. Saryčev, Index of second variation of a control system, Mat. Sb. (N.S.), 113(155) (1980), 464-486, 496. |
[29] |
L. F. Shampine, H. A. Watts and S. Davenport, Solving non-stiff ordinary differential equations-the state of the art, SIAM Rev., 18 (1976), 376-411. |
[30] |
V. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies," Academic Press, 1967. |
show all references
References:
[1] |
A. A. Agrachev and A. V. Sarychev, On abnormals extremals for Lagrange variational problems, J. Math. Systems Estim. Control, 8 (1998), 87-118. |
[2] |
E. L Allgower and K. Georg, "Numerical Continuation Methods. An Introduction," Springer Series in Computational Mathematics, 13, Springer-Verlag, Berlin, 1990. |
[3] |
V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," Second edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. |
[4] |
J. T. Betts and S. O. Erb, Optimal low thrust trajectories to the moon, SIAM J. Appl. Dyn. Syst., 2 (2003), 144-170.
doi: 10.1137/S1111111102409080. |
[5] |
C. Bischof, A. Carle, P. Kladem and A. Mauer, Adifor 2.0: Automatic Differentiation of Fortran 77 programs, IEEE Computational Science and Engineering, 3 (1996), 18-32.
doi: 10.1109/99.537089. |
[6] |
G. A. Bliss, "Lectures on the Calculus of Variations," University of Chicago Press, Chicago, Ill., 1946. |
[7] |
A. Bombrun, J. Chetboun and J.-B. Pomet, Transfert Terre-Lune en poussée faible par contrôle feedback - La mission SMART-1, (French) INRIA Research Report, 5955 (2006), 1-27. |
[8] |
B. Bonnard, J.-B. Caillau and G. Picot, Geometric and numerical techniques in optimal control of the two and three-body problems, Commun. Inf. Syst., 10 (2010), 239-278. |
[9] |
B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control, ESAIM Control Optim. and Calc. Var., 13 (2007), 207-236. |
[10] |
B. Bonnard, J.-B. Caillau and E. Trélat, COTCOT: Short reference manual, ENSEEIHT-IRIT Technical Report RT/APO/05/1, (2005), 1-15. |
[11] |
B. Bonnard and M. Chyba, "Singular Trajectories and Their Role in Control Theory," Math. and Applications, 40, Springer-Verlag, Berlin, 2003. |
[12] |
B. Bonnard, L. Faubourg and E. Trélat, "Mécanique Céleste et Contôle des Véhicules Spatiaux," Mathématiques & Applications (Berlin), 51, Springer-Verlag, Berlin, 2006. |
[13] |
B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, (French)[Theory of the singularities of the input/output mapping and optimality of singular trajectories in the minimal-time problem], Forum Math., 5 (1993), 111-159.
doi: 10.1515/form.1993.5.111. |
[14] |
B. Bonnard, N. Shcherbakova and D. Sugny, The smooth continuation method in optimal control with an application to quantum systems, ESAIM Control Optim. and Calc. Var., 17 (2011), 267-292.
doi: 10.1051/cocv/2010004. |
[15] |
J.-B. Caillau, "Contribution à l'Etude du Contrôle en Temps Minimal des Transferts Orbitaux," Ph.D thesis, Toulouse University, 2000. |
[16] |
J.-B. Caillau, B. Daoud and J. Gergaud, On some Riemannian aspects of two and three-body controlled problems, in "Recent Advances in Optimization and its Applications in Engineering," Springer, (2010), 205-224. |
[17] |
J.-B. Caillau, B. Daoud and J. Gergaud, Discrete and differential homotopy in circular restricted three-body control, to appear in AIMS Proceedings, 2010. |
[18] |
J. Gergaud and T. Haberkorn, Homotopy method for minimum consumption orbit transfer problem, ESAIM Control Optim. Calc. Var., 12 (2006), 294-310.
doi: 10.1051/cocv:2006003. |
[19] |
G. Gómez, S. Koon, M. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, Invariant manifolds, the spatial three-body problem ans space mission design, in "The Proceedings of AIAA/AAS Astrodynamics Specialist Meeting," Quebec City, Quebec, Canada, 2001. |
[20] |
M. Guerra and A. Sarychev, Existence and Lipschitzian regularity for relaxed minimizers, in "Mathematical Control Theory and Finance," Springer, Berlin, (2008), 231-250. |
[21] |
J. E. Marsden and S. D. Ross, New methods in celestial mechanics and mission design, Bull. Amer. Math. Soc. (N.S), 43 (2006), 43-73. |
[22] |
K. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem," Applied Mathematical Sciences, 90, Springer-Verlag, New York, 1992. |
[23] | |
[24] |
H. Pollard, "Mathematical Introduction to Celestial Mechanics," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. |
[25] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962. |
[26] |
G. Racca, B. H. Foing and M. Coradini, SMART-1: The first time of Europe to the moon, Earth, Moon and Planets, 85-86 (2001), 379-390. |
[27] |
G. Racca, et al., SMART-1 mission description and development status, Planetary and Space Science, 50 (2002), 1323-1337. |
[28] |
A. V. Saryčev, Index of second variation of a control system, Mat. Sb. (N.S.), 113(155) (1980), 464-486, 496. |
[29] |
L. F. Shampine, H. A. Watts and S. Davenport, Solving non-stiff ordinary differential equations-the state of the art, SIAM Rev., 18 (1976), 376-411. |
[30] |
V. Szebehely, "Theory of Orbits: The Restricted Problem of Three Bodies," Academic Press, 1967. |
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