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Orbital transfers: optimization methods and recent results
1.  Scuola di Ingegneria Aerospaziale  University of Rome "La Sapienza", via Salaria 851, 00138 Rome, Italy 
References:
[1] 
P. J. Angeline, Evolutionary optimization versus particle swarm optimization: philosophy and performance Differences, Evolutionary programming VII, Lecture Notes in Computer Science, 1447, Springer (1998), 601610. 
[2] 
R. B. Barrar, An analytic proof that the Hohmanntype transfer is the true minimum twoimpulse transfer, Astronautica Acta, IX (1963), 111. 
[3] 
R. H. Battin, An introduction to the mathematics and methods of astrodynamics, AIAA Education Series, AIAA, New York (1987), 529530. 
[4] 
D. J. Bell and D. H. Jacobson, "Singular Optimal Control Problems," Academic Press, Inc., London, 1975. 
[5] 
R. Bellman, "Dynamic Programming," Princeton University Press, Princeton, 1957. 
[6] 
C. R. Bessette and D. B. Spencer, Optimal space trajectory design: A heuristicbased approach, Advances in the Astronautical Sciences, 124 (2006), 16111628. 
[7] 
C. R. Bessette and D. B. Spencer, Identifying optimal interplanetary trajectories through a genetic approach, Paper AIAA 20066306, (2006). 
[8] 
J. T. Betts, Optimal interplanetary orbit transfers by direct transcription, Journal of the Astronautical Sciences, 42 (1994), 247326. 
[9] 
G. A. Bliss, "Lectures on the Calculus of Variations," University of Chicago Press, Chicago (1946), 108112. 
[10] 
K. R. Brown, E. F. Harrold and G. W. Johnson, Rapid optimization of multipleburn rocket flights, NASA CR1430 (1969). 
[11] 
R. G. Brusch and T. L. Vincent, Numerical implementation of a secondorder variational endpoint condition, AIAA Journal, 8 (1970), 22302235. doi: 10.2514/3.6092. 
[12] 
A. E. Bryson and Y. C. Ho, "Applied Optimal Control," Ginn and Company, Waltham, 1969. 
[13] 
A. Carlisle and G. Dozier, An OffTheShelf PSO, Proceedings of the Workshop on Particle Swarm Optimization, Indianapolis (2001). 
[14] 
P. Cicala, "An Engineering Approach to the Calculus of Variations," Levrotto & Bella, Torino, 1957. 
[15] 
M. Clerc, The swarm and the queen: Towards a deterministic and adaptive particle swarm optimization, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 1999), Washington (1999). doi: 10.1109/CEC.1999.785513. 
[16] 
A. R. Cockshott and B. E. Hartman, Improving the Fermentation medium for Echinocandin B production part II: particle swarm optimization, Process Biochemistry, 36 (2001), 661669. doi: 10.1016/S00329592(00)002612. 
[17] 
B. A. Conway, Optimal lowthrust interception of earthcrossing asteroids, Journal of Guidance, Control, and Dynamics, 20 (1997), 9951002. doi: 10.2514/2.4146. 
[18] 
V. CoverstoneCarroll and S. N. Williams, Optimal low thrust trajectories using differential inclusion concepts, Journal of the Astronautical Sciences, 42 (1994), 379393. 
[19] 
R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, Proceedings of the Sixth International Symposium on Micromachine and Human Science, Nagoya, (1995). doi: 10.1109/MHS.1995.494215. 
[20] 
R. C. Eberhart and Y. Shi, Comparison between genetic algorithms and particle swarm optimization, Evolutionary programming VII, Lecture Notes in Computer Science, Springer 1447 (1998), 611616. doi: 10.1007/BFb0040812. 
[21] 
R. C. Eberhart and Y. Shi, Comparing inertia weights and constriction factors in particle swarm optimization, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2000), La Jolla (2000). 
[22] 
R. C. Eberhart and Y. Shi, Particle swarm optimization: developments, applications, and resources, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2001), Seoul (2001). 
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T. N. Edelbaum, Some extensions of the Hohmann transfer maneuver, ARS Journal, 29 (1959), 864865. 
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T. N. Edelbaum, Propulsion requirements for controllable satellites, ARS Journal, 31 (1961), 10791089. 
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A. P. Engelbrecht, "Computational Intelligence. An Introduction," Wiley, Chichester, 2007. 
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P. J. Enright and B. A. Conway, Optimal finitethrust spacecraft trajectories using collocation and nonlinear programming, Journal of Guidance, Control, and Dynamics, 14 (1991), 981985. doi: 10.2514/3.20739. 
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P. J. Enright and B. A. Conway, Discrete approximations to optimal trajectories using direct transcription and nonlinear programming, Journal of Guidance, Control, and Dynamics, 15 (1992), 9941002. doi: 10.2514/3.20934. 
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P. C. Fourie and A. A. Groenwold, Particle swarms in topology optimization, Proceedings of the Fourth World Congress of Structural and Multidisciplinary Optimization, Liaoning Electronic Press, (2001), 17711776. 
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P. C. Fourie and A. A. Groenwold, The particle swarm optimization algorithm in size and shape optimization, Structural and Multidisciplinary Optimization, 23 (2002), 259267. 
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Y. Gao and and C. Kluever, Lowthrust interplanetary orbit transfer using hybrid trajectory optimization method with multiple shooting, Paper AIAA 20045088 (2004). 
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D. E. Goldberg, "Genetic Algorithms in Search, Optimization, and Machine Learning," Addison Wesley, Boston, 1989. 
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C. R. Hargraves and S. W. Paris, Direct trajectory optimization using nonlinear programming and collocation, Journal of Guidance, Control, and Dynamics, 10 (1987), 338342. doi: 10.2514/3.20223. 
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R. Hassan, B. Cohanim and O. de Weck, Comparison of particle swarm optimization and the genetic algorithm, Paper AIAA 20051897, (2005). 
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G. A. Hazelrigg, Globally optimal impulsive transfers via Green's Theorem, Journal of Guidance, Control, and Dynamics, 7 (1983), 462470. doi: 10.2514/3.19879. 
[35] 
A. L. Herman and B. A. Conway, Direct optimization using collocation based on highorder GaussLobatto quadrature rules, Journal of Guidance, Control, and Dynamics, 19 (1996), 592599. doi: 10.2514/3.21662. 
[36] 
A. L. Herman and B. A. Conway, Optimal lowthrust, earthmoon orbit transfer, Journal of Guidance, Control, and Dynamics, 21 (1998), 141147. doi: 10.2514/2.4210. 
[37] 
N. Higashi and H. Iba, Particle swarm optimization with Gaussian mutation, Proceedings of the IEEE Swarm Intelligence Symposium (SIS 2003), Indianapolis (2003). 
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F. B. Hildebrand, "Introduction to Numerical Analysis," Dover, New York, 1987. 
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R. F. Hoelker and R. Silber, The bielliptical transfer between coplanar circular orbits, Proceedings of the 4th Symposium on Ballistic Missiles and Space Technology, Pergamon Press, New York, 3 (1961), 164175. 
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W. Hohmann, Die Erreichbarkeit der Himmelskoerper, Oldenbourg, Munich (1925); also 
[41] 
X. Hu and R. Eberhart, Solving constrained nonlinear optimization problems with particle swarm optimization, Proceedings of the Sixth World Multiconference on Systemics, Cybernetics and Informatics (SCI 2002), Orlando (2002). 
[42] 
X. Hu, R. Eberhart and Y. Shi, Engineering optimization with particle swarm, Proceedings of the IEEE Swarm Intelligence Symposium (SIS 2003), Indianapolis (2003). 
[43] 
X. Hu, Y. Shi and R. Eberhart, Recent advances in particle swarm, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2004), Portland (2004). 
[44] 
M. R. Ilgen, Hybrid method for computing optimal low thrust OTV trajectories, Advances in the Astronautical Sciences, 87 (1994), 941958. 
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M. R. Ilgen, Hybrid method for computing optimal low thrust OTV trajectories, Advances in the Astronautical Sciences, 87 (1999), 941958. 
[46] 
A. B. Jenkin, Representative mission trade studies for lowthrust transfers to geosynchronous orbits, paper AIAA 20045086 (2004). 
[47] 
V. Kalivarapu and E. Winer, Implementation of digital pheromones in particle swarm optimization for constrained optimization problems, Paper AIAA 20081974 (2008). 
[48] 
J. A. Kechichian, Reformulation of Edelbaum's lowthrust transfer problem using optimal control theory, Journal of Guidance, Control, and Dynamics, 20 (1997), 988994. doi: 10.2514/2.4145. 
[49] 
J. A. Kechichian, Lowthrust eccentricityconstrained orbit raising, Journal of Spacecraft and Rockets, 35 (1998), 327335. doi: 10.2514/2.3330. 
[50] 
J. A. Kechichian, Optimal altitudeconstrained lowthrust transfer between inclined circular orbits, Journal of the Astronautical Sciences, 54 (2006), 485503. 
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J. Kennedy and R. Eberhart, Particle swarm optimization, Proceedings of the IEEE International Conference on Neural Networks, Piscataway, (1995). doi: 10.1109/ICNN.1995.488968. 
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J. Kennedy and R. Eberhart, "Swarm Intelligence," Academic Press, San Diego, 2001. 
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M. S. Khurana, H. Winarto and A. K. Sinha, Application of swarm approach and artificial neural networks for airfoil shape optimization, Paper AIAA 20085954, (2008). 
[54] 
S. Kitayama, K. Yamazaki and M. Arakawa, Adaptive range particle swarm optimization, Paper AIAA 20066912, (2006). 
[55] 
C. A. Kluever and B. L. Pierson, Optimal lowthrust, threedimensional earthmoon trajectories, Journal of Guidance, Control, and Dynamics, 18 (1995), 830837. doi: 10.2514/3.21466. 
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C. A. Kluever and S. R. Oleson, Direct approach for computing nearoptimal lowthrust earthorbit transfers, Journal of Spacecraft and Rockets, 35 (1998), 509515. doi: 10.2514/2.3360. 
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D. F. Lawden, "Optimal Trajectories for Space Navigation," Butterworths, London, 1963. 
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D. F. Lawden, Optimal intermediatethrust arcs in a gravitational field, Astronautica Acta, 8 (1962), 106123. 
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R. Mendes, R., J. Kennedy and J. Neves, The fully informed particle swarm: simpler, maybe better, IEEE Transactions on Evolutionary Computation, 8 (2004), 204210. doi: 10.1109/TEVC.2004.826074. 
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A. Miele, General variational theory of the flight paths of RocketPowered aircraft, missiles, and satellite carriers, Astronautica Acta, 4 (1958), 1121. 
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A. Miele, V. K. Basapur and W. Y. Lee, Optimal trajectories for aeroassisted, noncoplanar orbital transfer, Acta Astronautica, 15 (1987), 399411. doi: 10.1016/00945765(87)901767. 
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show all references
References:
[1] 
P. J. Angeline, Evolutionary optimization versus particle swarm optimization: philosophy and performance Differences, Evolutionary programming VII, Lecture Notes in Computer Science, 1447, Springer (1998), 601610. 
[2] 
R. B. Barrar, An analytic proof that the Hohmanntype transfer is the true minimum twoimpulse transfer, Astronautica Acta, IX (1963), 111. 
[3] 
R. H. Battin, An introduction to the mathematics and methods of astrodynamics, AIAA Education Series, AIAA, New York (1987), 529530. 
[4] 
D. J. Bell and D. H. Jacobson, "Singular Optimal Control Problems," Academic Press, Inc., London, 1975. 
[5] 
R. Bellman, "Dynamic Programming," Princeton University Press, Princeton, 1957. 
[6] 
C. R. Bessette and D. B. Spencer, Optimal space trajectory design: A heuristicbased approach, Advances in the Astronautical Sciences, 124 (2006), 16111628. 
[7] 
C. R. Bessette and D. B. Spencer, Identifying optimal interplanetary trajectories through a genetic approach, Paper AIAA 20066306, (2006). 
[8] 
J. T. Betts, Optimal interplanetary orbit transfers by direct transcription, Journal of the Astronautical Sciences, 42 (1994), 247326. 
[9] 
G. A. Bliss, "Lectures on the Calculus of Variations," University of Chicago Press, Chicago (1946), 108112. 
[10] 
K. R. Brown, E. F. Harrold and G. W. Johnson, Rapid optimization of multipleburn rocket flights, NASA CR1430 (1969). 
[11] 
R. G. Brusch and T. L. Vincent, Numerical implementation of a secondorder variational endpoint condition, AIAA Journal, 8 (1970), 22302235. doi: 10.2514/3.6092. 
[12] 
A. E. Bryson and Y. C. Ho, "Applied Optimal Control," Ginn and Company, Waltham, 1969. 
[13] 
A. Carlisle and G. Dozier, An OffTheShelf PSO, Proceedings of the Workshop on Particle Swarm Optimization, Indianapolis (2001). 
[14] 
P. Cicala, "An Engineering Approach to the Calculus of Variations," Levrotto & Bella, Torino, 1957. 
[15] 
M. Clerc, The swarm and the queen: Towards a deterministic and adaptive particle swarm optimization, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 1999), Washington (1999). doi: 10.1109/CEC.1999.785513. 
[16] 
A. R. Cockshott and B. E. Hartman, Improving the Fermentation medium for Echinocandin B production part II: particle swarm optimization, Process Biochemistry, 36 (2001), 661669. doi: 10.1016/S00329592(00)002612. 
[17] 
B. A. Conway, Optimal lowthrust interception of earthcrossing asteroids, Journal of Guidance, Control, and Dynamics, 20 (1997), 9951002. doi: 10.2514/2.4146. 
[18] 
V. CoverstoneCarroll and S. N. Williams, Optimal low thrust trajectories using differential inclusion concepts, Journal of the Astronautical Sciences, 42 (1994), 379393. 
[19] 
R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, Proceedings of the Sixth International Symposium on Micromachine and Human Science, Nagoya, (1995). doi: 10.1109/MHS.1995.494215. 
[20] 
R. C. Eberhart and Y. Shi, Comparison between genetic algorithms and particle swarm optimization, Evolutionary programming VII, Lecture Notes in Computer Science, Springer 1447 (1998), 611616. doi: 10.1007/BFb0040812. 
[21] 
R. C. Eberhart and Y. Shi, Comparing inertia weights and constriction factors in particle swarm optimization, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2000), La Jolla (2000). 
[22] 
R. C. Eberhart and Y. Shi, Particle swarm optimization: developments, applications, and resources, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2001), Seoul (2001). 
[23] 
T. N. Edelbaum, Some extensions of the Hohmann transfer maneuver, ARS Journal, 29 (1959), 864865. 
[24] 
T. N. Edelbaum, Propulsion requirements for controllable satellites, ARS Journal, 31 (1961), 10791089. 
[25] 
A. P. Engelbrecht, "Computational Intelligence. An Introduction," Wiley, Chichester, 2007. 
[26] 
P. J. Enright and B. A. Conway, Optimal finitethrust spacecraft trajectories using collocation and nonlinear programming, Journal of Guidance, Control, and Dynamics, 14 (1991), 981985. doi: 10.2514/3.20739. 
[27] 
P. J. Enright and B. A. Conway, Discrete approximations to optimal trajectories using direct transcription and nonlinear programming, Journal of Guidance, Control, and Dynamics, 15 (1992), 9941002. doi: 10.2514/3.20934. 
[28] 
P. C. Fourie and A. A. Groenwold, Particle swarms in topology optimization, Proceedings of the Fourth World Congress of Structural and Multidisciplinary Optimization, Liaoning Electronic Press, (2001), 17711776. 
[29] 
P. C. Fourie and A. A. Groenwold, The particle swarm optimization algorithm in size and shape optimization, Structural and Multidisciplinary Optimization, 23 (2002), 259267. 
[30] 
Y. Gao and and C. Kluever, Lowthrust interplanetary orbit transfer using hybrid trajectory optimization method with multiple shooting, Paper AIAA 20045088 (2004). 
[31] 
D. E. Goldberg, "Genetic Algorithms in Search, Optimization, and Machine Learning," Addison Wesley, Boston, 1989. 
[32] 
C. R. Hargraves and S. W. Paris, Direct trajectory optimization using nonlinear programming and collocation, Journal of Guidance, Control, and Dynamics, 10 (1987), 338342. doi: 10.2514/3.20223. 
[33] 
R. Hassan, B. Cohanim and O. de Weck, Comparison of particle swarm optimization and the genetic algorithm, Paper AIAA 20051897, (2005). 
[34] 
G. A. Hazelrigg, Globally optimal impulsive transfers via Green's Theorem, Journal of Guidance, Control, and Dynamics, 7 (1983), 462470. doi: 10.2514/3.19879. 
[35] 
A. L. Herman and B. A. Conway, Direct optimization using collocation based on highorder GaussLobatto quadrature rules, Journal of Guidance, Control, and Dynamics, 19 (1996), 592599. doi: 10.2514/3.21662. 
[36] 
A. L. Herman and B. A. Conway, Optimal lowthrust, earthmoon orbit transfer, Journal of Guidance, Control, and Dynamics, 21 (1998), 141147. doi: 10.2514/2.4210. 
[37] 
N. Higashi and H. Iba, Particle swarm optimization with Gaussian mutation, Proceedings of the IEEE Swarm Intelligence Symposium (SIS 2003), Indianapolis (2003). 
[38] 
F. B. Hildebrand, "Introduction to Numerical Analysis," Dover, New York, 1987. 
[39] 
R. F. Hoelker and R. Silber, The bielliptical transfer between coplanar circular orbits, Proceedings of the 4th Symposium on Ballistic Missiles and Space Technology, Pergamon Press, New York, 3 (1961), 164175. 
[40] 
W. Hohmann, Die Erreichbarkeit der Himmelskoerper, Oldenbourg, Munich (1925); also 
[41] 
X. Hu and R. Eberhart, Solving constrained nonlinear optimization problems with particle swarm optimization, Proceedings of the Sixth World Multiconference on Systemics, Cybernetics and Informatics (SCI 2002), Orlando (2002). 
[42] 
X. Hu, R. Eberhart and Y. Shi, Engineering optimization with particle swarm, Proceedings of the IEEE Swarm Intelligence Symposium (SIS 2003), Indianapolis (2003). 
[43] 
X. Hu, Y. Shi and R. Eberhart, Recent advances in particle swarm, Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2004), Portland (2004). 
[44] 
M. R. Ilgen, Hybrid method for computing optimal low thrust OTV trajectories, Advances in the Astronautical Sciences, 87 (1994), 941958. 
[45] 
M. R. Ilgen, Hybrid method for computing optimal low thrust OTV trajectories, Advances in the Astronautical Sciences, 87 (1999), 941958. 
[46] 
A. B. Jenkin, Representative mission trade studies for lowthrust transfers to geosynchronous orbits, paper AIAA 20045086 (2004). 
[47] 
V. Kalivarapu and E. Winer, Implementation of digital pheromones in particle swarm optimization for constrained optimization problems, Paper AIAA 20081974 (2008). 
[48] 
J. A. Kechichian, Reformulation of Edelbaum's lowthrust transfer problem using optimal control theory, Journal of Guidance, Control, and Dynamics, 20 (1997), 988994. doi: 10.2514/2.4145. 
[49] 
J. A. Kechichian, Lowthrust eccentricityconstrained orbit raising, Journal of Spacecraft and Rockets, 35 (1998), 327335. doi: 10.2514/2.3330. 
[50] 
J. A. Kechichian, Optimal altitudeconstrained lowthrust transfer between inclined circular orbits, Journal of the Astronautical Sciences, 54 (2006), 485503. 
[51] 
J. Kennedy and R. Eberhart, Particle swarm optimization, Proceedings of the IEEE International Conference on Neural Networks, Piscataway, (1995). doi: 10.1109/ICNN.1995.488968. 
[52] 
J. Kennedy and R. Eberhart, "Swarm Intelligence," Academic Press, San Diego, 2001. 
[53] 
M. S. Khurana, H. Winarto and A. K. Sinha, Application of swarm approach and artificial neural networks for airfoil shape optimization, Paper AIAA 20085954, (2008). 
[54] 
S. Kitayama, K. Yamazaki and M. Arakawa, Adaptive range particle swarm optimization, Paper AIAA 20066912, (2006). 
[55] 
C. A. Kluever and B. L. Pierson, Optimal lowthrust, threedimensional earthmoon trajectories, Journal of Guidance, Control, and Dynamics, 18 (1995), 830837. doi: 10.2514/3.21466. 
[56] 
C. A. Kluever and S. R. Oleson, Direct approach for computing nearoptimal lowthrust earthorbit transfers, Journal of Spacecraft and Rockets, 35 (1998), 509515. doi: 10.2514/2.3360. 
[57] 
R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals, AIAA Journal, 3 (1965), 14391444. doi: 10.2514/3.3165. 
[58] 
S. Koziel and Z. Michalewicz, Evolutionary algorithms, homorphous mappings, and constrained parameter optimization, Evolutionary Computation, 7 (1999), 1944. doi: 10.1162/evco.1999.7.1.19. 
[59] 
D. F. Lawden, "Optimal Trajectories for Space Navigation," Butterworths, London, 1963. 
[60] 
D. F. Lawden, Optimal intermediatethrust arcs in a gravitational field, Astronautica Acta, 8 (1962), 106123. 
[61] 
G. Leitmann, A calculus of variations solution of Goddard's problem, Astronautica Acta, 2 (1956), 5562. 
[62] 
G. Leitmann (Ed.), "Optimization Techniques," Academic Press, New York, 1962. 
[63] 
J. P. Marec, "Optimal Space Trajectories," Elsevier, Amsterdam, 1979. 
[64] 
S. McAdoo, D. J. Jezewski and G. S. Dawkins, Development of a method for optimal maneuver analysis of complex space missions, NASA TN D7882 (1975). 
[65] 
R. Mendes, R., J. Kennedy and J. Neves, The fully informed particle swarm: simpler, maybe better, IEEE Transactions on Evolutionary Computation, 8 (2004), 204210. doi: 10.1109/TEVC.2004.826074. 
[66] 
A. Miele, General variational theory of the flight paths of RocketPowered aircraft, missiles, and satellite carriers, Astronautica Acta, 4 (1958), 1121. 
[67] 
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