Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T22:21:14.652Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Space Trajectories—A Review of Published Work

Published online by Cambridge University Press:  04 July 2016

D. J. Bell
D. J. Bell*
Affiliation:
School of Mathematics, Bath University of Technology
Get access Rights & Permissions [Opens in a new window]

Extract

The problem of transferring a space vehicle between two points in a given gravitational field such that the minimum amount of fuel is used has been called the fundamental navigational problem of astronautics. In such a problem it may be required to find the optimum thrust magnitude and thrust direction which yields a minimum fuel trajectory. Furthermore, certain end conditions may be specified which the optimal trajectory must satisfy. In a large number of published papers the velocity of the vehicle is supposed known both at the beginning of the transfer and at the end whereas the time taken to complete the manoeuvre may or may not be given. Also, other performance criteria have been chosen besides minimum fuel. For example, minimum time of transit or maximum orbital altitude at perigee.

Papers mentioned in this review deal mainly with flight in two dimensions apart from those sections on general theory. Furthermore, all space vehicles considered are assumed to have a fixed exhaust velocity unless otherwise stated.

Type
Research Article
Information
The Aeronautical Journal , Volume 72 , Issue 686 , February 1968 , pp. 141 - 146
Copyright
Copyright © Royal Aeronautical Society 1968 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Now at The University of Manchester Institute of Science and Technology. Dept of Mathematics.

References

1. Goddard, R. H. A Method of Reaching Extreme Altitudes. Smithsonian Inst Publ Misc Collect 71, No 2, 1919.Google Scholar
2. Hohmann, W. Die Erreichbarkeit der Himmelskorper, Oldenbourg, Munich, 1925.Google Scholar
3. Bliss, G. A. Lectures on the Calculus of Variations— Part II. University of Chicago Press, 1946.Google Scholar
4. Lawden, D. F. General Motion of a Rocket in a Gravitational Field. JBIS, Vol 6, pp 187–191, December 1947.Google Scholar
5. Forbes, G. F. The Trajectory of a Powered Rocket in Space. IBIS, Vol 9, pp 75–79, March 1950.Google Scholar
6. Lawden, D. F. Note on a Paper by G. F. Forbes. JBIS, Vol 9, pp 230–234, September 1950.Google Scholar
7. Lawden, D. F. Minimal Trajectories. JBIS, Vol 9, pp 179–186, November 1950.Google Scholar
8. Lawden, D. F. Entry into Circular Orbits—I. JBIS, Vol 10, pp 5–17, January 1951.Google Scholar
9. Lawden, D. F. The Determination of Minimal Orbits. JBIS, Vol 11, pp 216–224, September 1952.Google Scholar
10. Lawden, D. F. Orbital Transfer via Tangential Ellipses. JBIS, Vol 11, pp 278–289, November 1952.Google Scholar
11. Lawden, D. F. Inter-Orbital Transfer of a Rocket. Annual Report of BIS, pp 321–333, 1952.Google Scholar
12. Lawden, D. F. Escape to Infinity from Circular Orbits. JBIS, Vol 12, pp 68–71, March 1953.Google Scholar
13. Tsien, H. S. Take-Off from Satellite Orbit. Jet Propulsion, Vol 23, pp 233–236, July-August 1953.Google Scholar
14. Lawden, D. F. Minimal Rocket Trajectories. JAmRS, Vol 23, pp 360–367, 382, November-December 1953.Google Scholar
15. Ehricke, K. A. Take-Off from Satellite Orbits. Jet Propulsion, Vol 23, pp 372–374, November-December 1953.Google Scholar
16. Lawden, D. F. Entry Into Circular Orbits—II. JBIS Vol 13, pp 27–32, January 1954.Google Scholar
17. Lawden, D. F. Fundamentals of Space Navigation. JBIS, Vol 13, pp 87–101, March 1954.Google Scholar
18. Lawden, D. F. Stationary Rocket Trajectories. j. Mech Appl Math, Vol 7, pp 488–504, 1954.Google Scholar
19. Lawden, D. F. Optimal Programming of Rocket Thrust ; Direction. Astronautica Acta, pp 41–56, 1955.Google Scholar
20. Lawden, D. F. Optimal Transfer Between Circular Orbits About Two Planets. Astronautica Acta, pp 89–99, 1955.Google Scholar
21. Lawden, D. F. Optimum Launching of a Rocket Into an Orbit About the Earth. Astronautica Acta, pp 185–190, 1955.Google Scholar
22. Lawden, D. F. Dynamic Problems of Interplanetary Flight. Aero Quart, Vol 6, pp 165–180, 1955.Google Scholar
23. Leitmann, G. A Calculus of Variations Solution of Goddard's Problem. Astronautica Acta, Vol 2, pp 55–62, 1956.Google Scholar
24. Lawden, D. F. Transfer Between Circular Orbits. Jet Propulsion, Vol 26, Part I, pp 555–558, July 1956.Google Scholar
25. Fried, B. D. and Richardson, J. M. Optimum Rocket Trajectories. J Appl Phys, Vol 27, pp 955–961, August 1956.Google Scholar
26. Fried, B. D. On the Powered Flight Trajectory of an Earth Satellite. Jet Propulsion, Vol 27, pp 641–643, June 1957.Google Scholar
27. Lawden, D. F. Mathematical Problems of Astronautics. Mathematical Gazette, Vol 41, pp 168–179, 1957.Google Scholar
28. Lawden, D. F. Optimal Rocket Trajectories. Jet Propulsion, Vol 27, p 1263, December 1957.Google Scholar
29. Peterson, J. B. Note on Take-Off from Satellite Orbit. Jet Propulsion, Vol 27, pp 1263–1264, December 1957.Google Scholar
30. Leitmann, G. A Note on Goddard's Problem. Astronautica Acta, Vol 3, pp 237–240, 1957.Google Scholar
31. Vargo, L. G. Criteria for Orbital Entry. Jet Propulsion, Vol 28, pp 54–55, January 1958.Google Scholar
32. Benney, D. J. Escape from a Circular Orbit Using Tangential Thrust. Jet Propulsion, Vol 28, pp 167–169, March 1958.Google Scholar
33. Dobrowolski, A. Satellite Orbit Perturbations Under a Continuous Radial Thrust of Small Magnitude. Jet Propulsion, Vol 28, pp 687–689, October 1958.Google Scholar
34. Leitman, G. Optimum Thrust Direction for Maximum Range. JBIS, Vol 16, pp 503–507, 1958.Google Scholar
35. Lawden, D. F. Optimal Escape from a Circular Orbit. Astronautica Acta, pp 218–233, 1958.Google Scholar
36. Hoelker, R. F. and Silber, R. The Bi-Elliptic Transfer Between Circular Co-Planar Orbits. Army Ballistic Missile Agency, Redstone Arsenal, Alabama, DA Tech Memo No 2-59, January 1959.Google Scholar
37. Perkins, F. M. Flight Mechanics of Low-Thrust Spacecraft. J Aero/Sp Sci, Vol 26, pp 291–297, May 1959.Google Scholar
38. Munick, H. An Optimum Transfer Path from an Elliptical Orbit to a Higher Energy Circular Orbit. JAmRS, Vol 29, pp 449–451, June 1959.Google Scholar
39. Breakwell, J. V. The Optimisation of Trajectories. J Soc Ind Appl Math, Vol 7, pp 215–247, 1959.Google Scholar
40. Long, R. S. Escape from a Circular Orbit with Finite Velocity at Infinity. Astronautica Acta, Vol 5, pp 159–162, 1959.Google Scholar
41. Smith, G. C. The Calculation of Minimal Orbits. Astronautica Acta, Vol 5, pp 253–265, 1959.Google Scholar
42. Rosa, S. Composite Trajectories Yielding Maximum Coasting Apogee Velocity. JAmRS, Vol 29, pp 843–848, 1959.Google Scholar
43. Edelbaum, T. N. Some Extensions of the Hohmann Transfer Maneuver. JAmRS, Vol 29, pp 864–865, 1959.Google Scholar
44. Leitmann, G. On a Class of Variational Problems in Rocket Flight. J Aero/Sp Sci, Vol 26, pp 586–591, September 1959.Google Scholar
45. Lawden, D. F. Necessary Conditions for Optimal Rocket Trajectories. J. Mech Appl Math, Vol 12, pp 476–487, 1959.Google Scholar
46. Lawden, D. F. Interplanetary Rocket Trajectories. Advances in Space Science, Vol 1, Chapter 1, Academic Press, New York, 1959.Google Scholar
47. Munick, H. and Mcgill, R. Analytical Estimates for Optimum Transfer Paths. JAmRS, Vol 30, pp 120–122, 1960.Google Scholar
48. Ting, L. Optimum Orbital Transfer by Several Impulses. Astronautica Acta, Vol 6, pp 256–265, 1960.Google Scholar
49. Moeckel, W. E. Trajectories with Constant Tangential Thrust in Central Gravitational Fields. NASA Lewis Research Center TR R-53, 1960.Google Scholar
50. Barrère, M. et al. Rocket Propulsion, pp 724–727, Amsterdam: Elsevier Publishing Company, 1960.Google Scholar
51. Ting, L. Optimum Orbital Transfer by Impulses. JAmRS, Vol 30, pp 1013–1018, 1960.Google Scholar
52. Mackay, J. S. Approximate Solution for Rocket Flight with Linear-Tangent Thrust Attitude Control. JAmRS, Vol 30, pp 1091–1093, 1960.Google Scholar
53. Moeckel, W. E. Trajectories with Constant Tangential Thrust in Central Gravitational Fields. NASA Techn Rep R-53, 1960.Google Scholar
54. Kalman, R. E. On the General Theory of Control Systems. Proceedings of the 1st International Congress on Automatic Control, Moscow, 1960. Butterworths Scientific Publications Ltd, London, Vol 1, p 481, 1961.Google Scholar
55. Ewing, G. M. A Fundamental Problem of Navigation in Free Space. Q of Appl Math, Vol 18, p 355, 1961.Google Scholar
56. Ting, L. Approximating Technique for Variational Problems. Aero Eng, Vol 20, pp 32–33, 89-91, 1961.Google Scholar
57. Faulders, C. R. Minimum-Time Steering Programs for Orbital Transfer with Low-Thrust Rockets. Astronautica Acta, Vol 7, pp 35–49, 1961.Google Scholar
58. Melbourne, W. G. Interplanetary Trajectories and Payload Capabilities of Advanced Propulsion Vehicles. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, Techn Rep No 32-28, March 1961.Google Scholar
59. Berkovitz, L. D. Variational Methods in Problems of Control and Programming. J Math Analysis and Applic, Vol 3, pp 145–169, 1961.Google Scholar
60. Rider, L. Characteristic Velocity Requirements for Impulsive Thrust Transfer Between Non-coplanar Circular Orbits. JAmRS, Vol 31, pp 345–351, 1961.Google Scholar
61. Moskowitz, S. E. and Ting, L. A Realistic Approach to Problems of Optimum Rocket Trajectories. JAmRS, Vol 31, pp 551–553, 1961.Google Scholar
62. Lawden, D. F. Optimal Powered Arcs in an Inverse Square Law Field. JAmRS, Vol 31, pp 566–568, 1961.Google Scholar
63. Ehlers, E. E. Missile Trajectories with Linear Time Variation of the Sine or Tangent of the Thrust Angle. JAmRS, Vol 31, pp 631–640, 1961.Google Scholar
64. Edelbaum, T. N. Propulsion Requirements for Controllable Satellites. JAmRS, Vol 31, pp 1079–1089, 1961.Google Scholar
65. Bryson, A. E. and Denham, W. F. Multivariable Terminal Control for Minimum Mean Square Deviation from a Nominal Path. Proceedings of the IAS Symposium on Vehicle Systems Optimization. Institute of Aerospace Sciences, New York, November 1961.Google Scholar
66. Lawden, D. F. Optimal Intermediate-Thrust Arcs in a Gravitational Field. Astronautica Acta, pp 106–123, 1962.Google Scholar
67. Contensou, P. Etude Théorique des Trajectories Optimales dans un Champ de Gravitation. Application au cas d'un Centre d'Attraction Unique. Astronautica Acta, Vol 8, pp 134–150, 1962.Google Scholar
68. Bryson, A. E. and Denham, W. F. A Steepest-ascent Method for Solving Optimum Programming Problems. J. Appl Mech, Vol 29, pp 247–257, 1962.Google Scholar
69. Kelley, H. J. Guidance Theory and Extremal Fields. Inst Radio Engrs Auto Control, Vol 7, pp 75–82, October 1962.Google Scholar
70. Lawden, D. F. Impulsive Transfer Between Elliptical Orbits. Optimization Techniques, Chapter 11, Academic Press, New York, 1962.Google Scholar
71. Leitmann, G. (Editor). Optimization Techniques with Applications to Aerospace Systems. Academic Press Inc, New York, 1962.Google Scholar
72. Zee, C. H. Effect of Finite Thrusting Time in Orbital Maneuvers. JAIAA, Vol 1, pp 60–64, 1963.Google Scholar
73. Barrar, R. B. Two-Impulse Transfer vs One-Impulse Transfer: Analytic Theory. JAIAA, Vol 1, pp 65–68, 1963.Google Scholar
74. Barrar, R. B. An Analytic Proof that the Hohmann-type Transfer is the True Minimum Two-impulse Transfer. Astronautica Acta, Vol 9, pp 1–11, 1963.Google Scholar
75. Moskowitz, S. E. On the Accuracy of Approximate Steering Schedules in Optimal Correctional Maneuvers. Astronautica Acta, Vol 9, pp 20–30, 1963.Google Scholar
76. Bell, D. J. A Review of Flight Optimisation Along Synergic Paths in the Period 1945-1960. Journal of the Royal Aeronautical Society, Vol 67, pp 119–123, February 1963.Google Scholar
77. Ling, and M., Pierucci Optimum Launching of a Satellite by Two Impulses of Unequal Specific Impulse. Astronautica Acta, Vol 9, pp 174–183, 1963.Google Scholar
78. Zee, C. H. Low Thrust Oscillatory Spiral Trajectory. Astronautica Acta, Vol 9, pp 201–207, 1963.Google Scholar
79. Lawden, D. F. Analytical Techniques for the Optimisation of Rocket Trajectories. Aero Quart, Vol 14, pp 105–124, May 1963.Google Scholar
80. Breakwell, J. V., Speyer, J. L. and Bryson, A. E. Optimisation and Control of Non-linear Systems Using the Second Variation. J Soc Indus App Math Control, Vol 1, pp 193–223, 1963.Google Scholar
81. Kelley, H. J. Singular Extremals in Lawden's Problem of Optimal Rocket Flight. JAIAA, Vol 1, pp 1578–1580, 1963.Google Scholar
82. Zee, C. H. Low Constant Tangential Thrust Spiral Trajectories. JAIAA, Vol 1, pp 1581–1583, 1963.Google Scholar
83. Mccue, G. A. Optimum Two-impulse Orbital Transfer and Rendezvous Between Inclined Elliptical Orbits. JAIAA, Vol 1, pp 1865–1872, 1963.Google Scholar
84. Bryson, A. E., Denham, W. F. and Dreyfus, S. E. Optimal Programming Problems with Inequality Constraints I: Necessary Conditions for Extremal Solutions. JAIAA, Vol 1, pp 2544–2550, 1963.Google Scholar
85. Lawden, D. F. Optimal Trajectories for Space Navigation. Butterworths Mathematical Texts, 1963.Google Scholar
86. Denham, W. F. and Bryson, A. E. Optimal Programming Problems with Inequality Constraints II: Solution by Steepest Ascent. JAIAA, Vol 2, pp 25–35, 1964.Google Scholar
87. Mccue, G. A. Optimization and Visualization of Functions. JAIAA, Vol 2, pp 99–100, 1964.Google Scholar
88. Jurovics, S. A. Orbital Transfer by Optimum Thrust Direction and Duration. North American Aviation Inc, SID 64-29, February 1964.Google Scholar
89. Edelbaum, T. M. Optimum Low Thrust Rendezvous and Station Keeping. JAIAA, Vol 2, pp 1196–1201, 1964.Google Scholar
90. Pfeiffer, C. G. A Geometrical Interpretation of an Optimal Trajectory. California Institute of Technology, Jet Propulsion Laboratory, Space Programs Summary 37-27, Vol IV, July 1964.Google Scholar
91. Lubard, S. Optimum Launching to Hyperbolic Orbit by Two Impulses of Unequal Specific Impulse. Astronautica Acta, Vol 10, pp 138–151, 1964.Google Scholar
92. Jazwinski, A. H. Optimal Trajectories and Linear Control of Nonlinear Systems. JAIAA, Vol 2, p 1371, 1964.Google Scholar
93. Kelley, H. J. A Second Variation Test for Singular Extremals. JAIAA, Vol 2, pp 1380–1382, 1964.Google Scholar
94. Pfeiffer, C. G. A Fundamental Theorem on Optimal Control. California Institute of Technology. Jet Propulsion Laboratory, Space Programs Summary 37-28, Vol IV, August 1964.Google Scholar
95. Mcgill, R. and Kenneth, P. Solution of Variational Problems by Means of a Generalized Newton-Raphson Operator. JAIAA, Vol 2, pp 1761–1766, 1964.Google Scholar
96. Lee, G. An Analysis of Two-impulse Orbital Transfer. JAIAA, Vol 2, pp 1767–1773, 1964.Google Scholar
97. Vinh, N. X. A Property of Cotangential Elliptical Transfer Orbits. JAIAA, Vol 2, pp 1841–1844, 1964.Google Scholar
98. Keller, J. L. On Minimum Propellant Paths for Thrust Limited Rockets. Astronautica Acta, Vol 10, pp 262–269, 1964.Google Scholar
99. Lawden, G. H. Trajectory Optimization for a Rocket with a Generalized Thrust Characteristic. Astronautica Acta, Vol 10, pp 279–295, 1964.Google Scholar
100. Pines, S. Constants of the Motion for Optimal Thrust Trajectories in a Central Force Field. JAIAA, Vol 2, pp 2010–2014, 1964.Google Scholar
101. Ting, L. and Brofman, S. On Take-Off from Circular Orbit by Small Thrust. Z Angew Math Mech, Vol 44, pp 417–428, October/November 1964.Google Scholar
102. Pfeiffer, C. G. Some Theoretical Considerations Arising in Guidance Analysis. American Astronautical Society; American Association for the Advancement of Science. Paper 3, December 1964.Google Scholar
103. Battin, R. H. Astronautical Guidance, Chapter 10, pp 341–374, McGraw-Hill Book Co Inc, New York, 1964.Google Scholar
104. Hillsley, R. H. and Robbins, H. M. A Steepest Ascent Trajectory Optimization Method which Reduces Memory Requirements. Computing Methods in Optimization Problems. Academic Press Inc, New York, pp 107–133, 1964.Google Scholar
105. Breakwell, J. V. Minimum Impulse Transfer. AIAA Progress in Astronautics and Aeronautics: Celestial Mechanics and Astrodynamics, edited by Saebehely, V. G., Academic Press Inc, New York, Vol 14, pp 583–589, 1964.Google Scholar
106. De Veubeke, B. Fraeijs and Geerts. Optimisation of Multiple-Impulse Orbital Transfers by the Maximum Principle. Proceedings of the XVth IAF Congress, Warsaw, Vol 1, pp 589–616, 1964.Google Scholar
107. Balakrishnan, A. V. and Neustadt, C. W. (Editors). Computing Methods in Optimization Problems. Academic Press Inc, New York, 1964.Google Scholar
108. Roth, H. L. Preliminary Investigations Relative to Multiple Rendezvous Between Circular Orbits. Aero-space Corp Report TDR 269 (4130-10)-4, 1964.Google Scholar
109. Cavoti, C. R. Method of Characteristic Constants and Retro-Thrust Optimality in a Gravitational Field—Necessary and Sufficient Conditions. J Franklin Inst, Vol 279, pp 169–188, March 1965.Google Scholar
110. Breakwell, J. V. A Doubly Singular Problem in Optimal Interplanetary Guidance. J Soc Ind Appl Math Control, Vol 3, pp 71–77, 1965.Google Scholar
111. Moyer, H. G. Minimum Impulse Coplanar Circle-ellipse Transfer. JAIAA, Vol 3, pp 723–726, 1965.Google Scholar
112. Tschauner, J. and Hempel, P. Rendezvous zu einem in elliptischer Bahn umlaufenden Ziel. Astronautica Acta, Vol 11, pp 104–109, 1965.Google Scholar
113. Busemann, A. Minimalprobleme der Luft-und Raumfahrt. The 1965 Ludwig Prandtl Memorial Lecture, Vienna (April 1965); Z fur Flugwiss, Vol 13, pp 401–411, 1965.Google Scholar
114. Mccue, G. A. and Bender, D. F. Optimum Transfer Between Nearly Tangent Orbits. North American Aviation Inc, SID 64-1097, May 1965.Google Scholar
115. Edelbaum, T. N. Optimum Power-Limited Orbit Transfer in Strong Gravity Fields. JAIAA, Vol 3, pp 921–925, 1965.Google Scholar
116. Robbins, H. M. Optimality of Intermediate-Thrust Arcs of Rocket Trajectories. JAIAA, Vol 3, pp 1094–1098, 1965.Google Scholar
117. Eckenwiler, M. W. Closed-Form Lagrangian Multipliers for Coast Periods of Optimum Trajectories. JAIAA, Vol 3, pp 1149–1151, 1965.Google Scholar
118. Tschauner, J. and Hempel, P. On Minimum-Fuel Rendezvous Techniques. AIAA Paper 65-65, 1965; also J Spacecraft and Rockets, Vol 2, pp 802–804, 1965.Google Scholar
119. Long, R. S. Newton-Raphson Operator; Problems With Undetermined End Points. JAIAA, Vol 3, p 1351, 1965.Google Scholar
120. Pierucci, M. Optimum Orbital Transfer by Two Impulses of Unequal Specific Impulse. Astronautica Acta, Vol 11, pp 268–270, 1965.Google Scholar
121. De veubeke, B. FRAEIJS. Canonical Transformations and the Thrust-Coast-Thrust Optimal Transfer Problem. Astronautica Acta, Vol 11, pp 271–282, 1965.Google Scholar
122. Kelley, H. J. A Transformation Approach to Singular Subarcs in Optimal Trajectory and Control Problems. J Soc Ind Appl Math, Control Vol 2, pp 234–240, 1965.Google Scholar
123. Kopp, R. E. and Moyer, H. G. Necessary Conditions for Singular Extremals. JAIAA, Vol 3, pp 1439–1444, August 1965.Google Scholar
124. Bell, D. J. Optimal Trajectories and the Accessory Minimum Problem. Aero Quart, Vol 16, pp 205–220, August 1965.Google Scholar
125. Anthony, M. L. and Sasaki, F. T. Rendezvous Problem for Nearly Circular Orbits. JAIAA, Vol 3, pp 1666–1673, 1965.Google Scholar
126. Marchal, C. Transferts optimaux entre orbites elliptiques coplanaires (Durée indifferente). Astronautica Acta, Vol 11, pp 432–445, 1965.Google Scholar
127. Cohen, M. J. Low Thrust Spiral Trajectories of a Satellite of Variable Mass. JAIAA, Vol 3, pp 1946–1949, 1965.Google Scholar
128. Paiewonsky, B. Optimal Control: A Review of Theory and Practice. JAIAA, Vol 3, pp 1985–2006, 1965.Google Scholar
129. Mccue, G. A. and Bender, D. F. Numerical Investigation of Minimum Impulse Orbital Transfer. JAIAA, Vol 3, pp 2328–2334, 1965.Google Scholar
130. Marec, J. P. Transferts economiques entre orbites infiniement proches. Proceedings of the XVI Congress of the International Astronautical Federation, 1965.Google Scholar
131. Edelbaum, T. N. HOW Many Impulses? AIAA Paper 66-7, 1966.Google Scholar
132. Mcintyre, J. E. Neighbouring Optimal Terminal Control with Discontinuous Forcing Functions. JAIAA, Vol 4, pp 141–148, January 1966.Google Scholar
133. Pfeiffer, C. G. A Technique for Optimal Final Value Control of Powered Flight Trajectories. JAIAA, Vol 4, pp 352–360, February 1966.Google Scholar
134. Shi, Y. Y. and Eckstein, M. C. Ascent or Descent from Satellite Orbit by Low Thrust. Douglas Aircraft Co Papei 3856, February 1966; also JAIAA, Vol 4, pp 2203–2209, 1966.Google Scholar
135. Denham, W. F. On Numerical Optimization with State Variable Inequality Constraints. JAIAA, Vol 4, pp 550–552, March 1966.Google Scholar
136. Breakwell, J. V. and Rauch, H. E. Optimum Guidance for a Low Thrust Interplanetary Vehicle. JAIAA, pp 693–704, April 1966.Google Scholar
137. Hempel, P. R. Representation of the Lagrangian Multipliers for Coast Periods of Optimum Trajectories. JAIAA, Vol 4, pp 729–730, April 1966.Google Scholar
138. Goh, B. S. The Second Variation for the Singular Bolza Problem. J Soc Ind Appl Math Control, Vol 4, pp 309–325, 1966.Google Scholar
139. Mcintyre, J. E. and Crocco, L. Linearized Treatment of Optimal Transfer of a Thrust-Limited Vehicle Between Coplanar Circular Orbits. Astronautica Acta, May-June 1966.Google Scholar
140. Handelsman, M. Optimal Free-Space Fixed-Thrust Trajectories Using Impulsive Trajectories as Starting Iteratives. JAIAA, Vol 4, pp 1077–1082, June 1966.Google Scholar
141. Moyer, H. G. Necessary Conditions for Optimal Single Impulse Transfer. JAIAA, Vol 4, pp 1405–1410, August 1966.Google Scholar
142. Robbins, H. M. An Analytical Study of the Impulsive Approximation. JAIAA, Vol 4, pp 1417–1423, 1966.Google Scholar
143. Edelbaum, T. N. An Asymptotic Solution for Optimum Power-Limited Orbit Transfer. JAIAA, Vol 4, pp 1491–1494, August 1966.Google Scholar
144. Schulman, and Scott, . Terminal Rendezvous for Elliptical Orbits. AIAA Preprint 66-533, 1966.Google Scholar
145. Walsh, G. R. The Fourth Necessary Condition in an Optimal Rocket Trajectory. J Inst Maths Applies, Vol 2, pp 237–247, September 1966.Google Scholar
146. Tapley, B. D. and Fowler, W. T. Terminal Guidance for Continuous Powered Space Vehicles. JAIAA, Vol 4, pp 1683–1684, September 1966.Google Scholar
147. Martin, F. H. Closed-Loop Near-Optimum Steering for a Class of Space Missions. JAIAA, Vol 4, pp 1920–1927, November 1966.Google Scholar
148. Goh, B. S. Necessary Conditions for Singular Extremals Involving Multiple Control Variables. J Soc Ind Appl Math Control, Vol 4, pp 716–731, 1966.Google Scholar
149. Kenneth, P. and Taylor, G. E. Solution of Variational Problems with Bounded Control Variables by Means of the Generalized Newton-Raphson Method. Recent Advances in Optimization Techniques, edited by Lovi, A. and Vogel., L. P. John Wiley & Sons Inc, New York, 1966.Google Scholar
150. Shi, Y. Y. and Eckstein, M. C. An Approximate Solution for Ascending Low-Thrust Trajectories Without Singularity. JAIAA, Vol 5, pp 170–172, January 1967.Google Scholar
151. Mcintyre, J. E. and Crocco, L. Higher Order Treatment of the Optimal Transfer of a Thrust-Limited Vehicle Between Coplanar Circular Orbits. Astronautica Acta, Vol 13, pp 3–21, January-February 1967.Google Scholar
152. Billik, B. H. and Roth, H. L. Studies Relative to Rendezvous Between Circular Orbits. Astronautica Acta, Vol 13, pp 23–36, 1967.Google Scholar
153. Culp, R. D. Contensou-Busemann Conditions for Optimal Coplanar Orbit Transfer. JAIAA, Vol 5, pp 371–372, February 1967.Google Scholar
154. Roth, H. L. Minimization of the Velocity Increment for a Bi-Elliptic Transfer with Plane Change. Astronautica Acta, Vol 13, pp 119–130, March-April 1967.Google Scholar
155. Mccue, G. A. Quasilinearization Determination of Optimum Finite-Thrust Orbital Transfers. JAIAA, Vol 5, pp 755–763, April 1967.Google Scholar
156. Anderson, G. M., Falb, P. L. and Robinson, A. C. One-Dimensional Minimum-Time Rendezvous for a Thrust-Limited Rocket. JAIAA, Vol 5, pp 1017–1019, May 1967.Google Scholar
157. Euler, E. A. and Schulman, Y. Second-Order Solution to the Elliptical Rendezvous Problem. JAIAA, Vol 5, pp 1033–1035, May 1967.Google Scholar
158. Tschauner, J. Elliptic Orbit Rendezvous. JAIAA, Vol 5, pp 1110–1113, June 1967.Google Scholar
159. Hempel, P. Ausdehnung des Hohmann-Ubergangs auf nicht-koaxiale Elliptische Bahnen. Astronautica Acta, pp 205–212, May-June 1967.Google Scholar
160. Marec, J. P. Transferts Economiques Plans de Type Hohmann Entre Orbites Quasi-Circulaires Coaxiales Proches. Astronautica Acta, pp 269–279, May-June 1967.Google Scholar