Skip to main content
Log in

Singular Optimal Controls of Rocket Motion (Survey)

  • Published:
International Applied Mechanics Aims and scope

Survey of modern state and discussion of problems of the perfection of methods of investigation of variational problems with a focus on mechanics of space flight are presented. The main attention is paid to the enhancement of the methods of solving of variational problems of rocket motion in the gravitational fields, including rocket motion in the atmosphere. These problems are directly connected with the permanently actual problem of the practical astronautics to increase the payload that is orbited by the carrier rockets in the circumplanetary orbits. An analysis of modern approaches to solving the problems of control of rockets and spacecraft motion on the trajectories with singular arcs that are optimal for the motion of the variable mass body in the medium with resistance is given. The presented results for some maneuvers can serve as an information source for decision making on designing promising rocket and space technology

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. V. N. Akimov, V. G. Konyukhov, and A. A Koroteev, “Efficiency of the use of space multimodal nuclear energy propulsion systems with a machine energy conversion,” Izv. RAN, Energetika, No. 3, 20–27 (2008).

  2. V. E. Alemasov, A. F. Dregalin, and A. P. Tishin, Theory of Rockets Engines [in Russian], Mashinostroenie, Moscow (1980).

    Google Scholar 

  3. M. Barrere, A. Jaumotte, B. F. Veubeke, and J. Vandenkerckhove, Rocket Engines [Russian translation], Oborongiz, Moscow (1962).

    Google Scholar 

  4. V. V. Beletsky, Sketches on Motions of Space Bodies [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  5. E. N. Belyaev, A. G. Vorobyov and E. M. Gnesin, “Development of mathematical model of the rocket engine, operating in the stationary mode,” Trudy MAI, No. 73, 16–33 (2014).

  6. V. I. Borzov, “Problem of motion separation in the flight dynamics,” Mechanics of Solids, No. 5, 3–11 (1981).

  7. D. F. Brekwell, “Optimization of trajectories,” Vopr. Raketn. Tekhniki, No. 1, 46–69 (1961).

  8. R. Gabasov and F. M. Kirillova, Singular Optimal Controls [in Russian], Nauka, Moscow (1973).

    MATH  Google Scholar 

  9. F. R Gantmakher and L. M. Levin, Flight Theory of Uncontrolled Rockets [in Russian], Gos. Izd. Fiz.-Mat. Lit., Moscow (1959).

    Google Scholar 

  10. I. N. Glukhikh, V. F. Chelyaev, and A. N. Shcherbakov, “Water use for protection of inhabited space objects from high-speed particles: micrometeors, space waste,” Izv. RAN, Energetika, No. 1, 92–95 (2009).

  11. E. A. Grebennikov and V. G. Demin, Interplanetary Flights [in Russian], Nauka, Moscow, (1965).

    Google Scholar 

  12. I. S. Grigoriev and K. G. Grigoriev, “To the problem of solving in impulse statement the problems of trajectory optimization for spacecraft flight with the jet engine of great thrust in the arbitrary gravitational field in a vacuum,” Cosmic Research, 40, No. 1, 88–111 (2002).

    Article  ADS  Google Scholar 

  13. I. S. Grigoriev and K. G. Grigoriev “On the use of solutions of the problems of the SC trajectories optimization of the impulse statement for solving the problem of optimal control of the SC trajectories with the engines with limited thrust,” Cosmic Research, No. 4, 358–366 (2007).

  14. G. L. Grozdovskii, Yu. N. Ivanov, and V.V. Tokarev, Mechanics of Space Flight. Problems of Optimization [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  15. G. L. Grozdovskiy, B. N. Kiforenko, and V. V. Kuzmenko, “Effect of weight restrictions on the maximal height ascent on motion in the atmosphere,” Uchon. Zap. TsAGI, 5, No. 1, 53–59 (1974).

    ADS  Google Scholar 

  16. G. L. Grozdovskii and B. N. Kiforenko, “Optimal sectioning of a material point of variable mass,” Vychisl Prikl. Matem., No. 18, 17–24 (1972).

  17. N. M. Dron’, A. I. Kondratiev, P. G. Khorolskii, and L. G. Dubovik, “Comparative estimation of characteristics of space trawlers for three variants of maneuver of their injection,” Aviation and Space Engineering and Technology, No. 10(77), 21–23 (2010).

  18. N. M. Dron’, L. G. Dubovik, A. I. Kondratiev and P. G. Khorolskiy, “Mass characteristics of space waste collecting apparatus, injected by known launch vehicle with the use of electric rocket propulsion system,” Mekh. Mashinostr., No. 1, 8–12 (2010).

  19. N. M. Dron’, P. G. Khorolskii, A. V. Khitko, and L. G. Dubovik, “Selection of an engine for space waste collector maneuvering at the stage of clearing the near-Earth space,” Mekh. Mashinostr., Prikl. Mekh., No. 2, 3–8 (2012).

  20. N. E. Zhukovskii, On the Theory of Vessels Driven by the Reaction of Outflowing Jet [in Russian], V. IV, ONTI NKTP SSSR, Moscow-Leningrad (1937).

  21. V. A. Zadontsev, “Two lives and two rockets of Werner von Braun (1912 – 1977). On the centenary of his birth,” Aviats. Kosmich. Tekh. Tekhnol., No. 9(96), 146–158 (2012).

  22. O. N. Zasukhin, P. V. Bulat, and N. V. Prodan, “Development of methods for calculating the base pressure,” Fundament. Issled., No. 6, 273–279 (2012).

  23. V. T. Zlatskii, Investigation of Degenerated Problems of Flight Mechanics [in Russian], Ph.D thesis (Phys&Math), Kiev (1982).

  24. V. T. Zlatskii and B. N. Kiforenko, “On computation of optimal trajectories with sections of singular control,” Vych. Pprikl. Mat., No. 49, 55–62 (1983).

  25. V. T. Zlatskii and B. N. Kiforenko, “Optimal operational speed on vertical motion of a body of variable mass in the atmosphere,” Trudy TsAGI, No. 1729, 7–13 (1976).

  26. V. V. Ivashkin and V. V. Smirnov, “Qualitative analyze of the certain methods for reduction of asteroid danger for the Earth,” Solar System Research, 27, No. 6, 46–54 (1993).

    Google Scholar 

  27. V. A. Il’in and G. E. Kuzmak, Optimum Spacecraft Flights [in Russian], Nauka, Moskow (1976).

    Google Scholar 

  28. A. Yu. Ishlinskii, “Two remarks on the theory of rockets motion,” Dokl. AN SSSR, 53, No. 7, 599–610 (1946).

    Google Scholar 

  29. B. N. Kiforenko, “To the question pf optimal control the rocket thrust magnitude in the atmosphere,” Mechanics of Solids, No. 3, 21–27 (1982).

  30. B. N. Kiforenko, “On the use of aerostatic lifting force on accumulation of the maximal energy by a body of variable mass,” Trudy TsAGI, No. 1729, 3–6 (1976).

  31. B. N. Kiforenko, “On optimization of parameters of a body of variable mass on motion with active waste discharge of the life support system,” Cosmic Research, 13, No. 2, 201–205 (1975).

    ADS  Google Scholar 

  32. B. N. Kiforenko, “On singular control in flight mechanics with limited power and energy accumulation,” Cosmic Research, 9, No. 4, 536–540 (1971).

    Google Scholar 

  33. B. N. Kiforenko, “Optimal trajectories with sections of singular control,” Slozhn. Sist. Upravl., Inst. Kibernetiki UN USSR, Kiev, 45–55 (1974).

  34. B. N. Kiforenko and G. K. Bagnyuk, “The use of aerostatic lifting force on motion of a body of variable mass in the dense atmosphere,” in: Proc. 20th Readings on Development of the Scientific Heritage and Ideas of K. E. Tsiolkovsky, Sec.: Problems of Rocket and Space Engineering [in Russian], IIET, Moscow (1987), pp. 21–28.

  35. B. N. Kiforenko and I. Yu. Vasil’ev, “Problems of optimization of manned interplanetary expeditions,” Kosmich. Nauka Tekhnol., 6, No. 1, 3–55 (2000).

    Article  ADS  Google Scholar 

  36. B. N. Kiforenko and G. K. Dauletov, “Analytical investigation of the optimal control of the working medium composition,” in: Proc. 9th Readings from S. P. Korolv [in Russian], Moscow (1987), pp. 100–109.

  37. B. N. Kiforenko, V. T. Zlatakii, and N. N. Kozhukhovskii, “Peculiarities of the optimal control of the the rocket thrust in the dense atmosphere,” in: Proc. 11th Readings on Development of the Scientific Heritage and Ideas of K. E. Tsiolkovsky, Sec.: Problems of Rocket and Space Engineering [in Russian], IIET, Moscow (1978), pp. 132–147.

  38. B. N. Kiforenko, V. T. Zlatakiy and V. V. Kuzmenko, “On the efficiency of the optimal control of the degree of gas extension in the rocket engine nozzle," in: Proc. 12th Readings on Development of the Scientific Heritage and Ideas of K. E. Tsiolkovsky, Sec.: Problems of Rocket and Space Engineering [in Russian], IIET, Moscow (1979), pp. 61–64.

  39. B. N. Kiforenko and V. V. Kuzmenko, “On the effect of external forces on motion of a material point of variable mass,” in: Proc. 8th Readings on Development of the Scientific Heritage and Ideas of K. E. Tsiolkovsky, Sec.: Problems of Rocket and Space Engineering [in Russian], IIET, Moscow (1975), pp. 114–119.

  40. B. N. Kiforenko and A. P. Sidorchuk. “Optimal control of the rocket engine with quasi-rated engine,” in: Scientific Work of K. E. Tsiolkovsky and Modern Development of His Ideas [in Russian], Nauka, Moscow (1984), pp. 35–37.

  41. B. N. Kiforenko and A. M. Kharitonov, “Mathematical modeling of the controlled dynamical objects,” Problem. Upravl. Inform., No. 4, 35–48 (2000).

  42. B. N. Kiforenko and A. M. Kharitonov, “Optimal control of fuel components consumption in accidental situations,” Probl. Upravl. Inform., No. 3, 25–33 (1988).

  43. B. N. Kiforenko and E. A. Khasenov, “Computation of the optimal trajectories of motion of a body of variable mass in the atmosphere,” Vychisl. Prikl. Matem., No. 51, 98–103 (1983).

  44. B. N. Kiforenko and E. A. Khasenov, “Optimal plane motions of a material point of variable mass,” in: Proc. 16th Readings on Development of the Scientific Heritage and Ideas of K. E. Tsiolkovsky, Sec.: Problems of Rocket and Space Engineering [in Russian], IIET, Moscow (1982), pp. 22–28.

  45. B. N. Kiforenko and E. A. Khasenov, “Stepwise thrust controlling in the problem of height ascending,” in: Proc. 8th Space Science Readings from S. P. Korolev [in Russian], Moscow (1986), pp. 119–125.

  46. M. S. Konstantinov, Kh. V. Leb, V. G. Petukhov, and G. A. Popov, “Design-ballistic analysis of a Mars manned mission with a nuclear electrorocket propulsion system,” Trudy MAI, No. 42, 21 p. (2011).

  47. A. A. Kosmodemyanskii, “General theorems of mechanics of bodies of variable mass,” Trudy VVIA im. N. E. Zhukovskogo, No. 184, 16 (1946).

  48. A. A. Kosmodemyanskii, “Extremal problems of mechanics of a point of variable mass,” Dokl. AN SSSR, 53, No. 1, 17–19 (1946).

    Google Scholar 

  49. Cosmonautics in the 21th Century. Report by V. A. Lopota, president of the S. P. Korolev RSC “Energiya,” Vest. RAN, 81, No. 9, 771–793 (2011).

  50. V. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  51. V. F. Krotov and M. M. Khrustalev, “Optimal control by thrust and attack angle of a flying vehicle on the maneuver ascending - acceleration,” in: Problems of Analytical Mechanics, Stability and Control Theory [in Russian], Nauka, Moscow (1975), pp. 168–174.

  52. A. N. Krylov, Lectures on Approximate Calculations [in Russian], Gos. Izdat. Techn.-Teoret. Liter., Moscow (1954).

    Google Scholar 

  53. V. V. Kuzmenko, “Singular control of the rocket motion in the atmosphere,” in: Proc. 9th Readings on Development of the Scientific Heritage and Ideas of K. E. Tsiolkovsky, Sec.: Problems of Rocket and Space Engineering [in Russian], IIET, Moscow (1975), pp. 138–146.

  54. I. V. Lomakin, M. B. Martynov, V. G. Pol’, and A. V. Simonov, “Asteroid danger, real problems and practical actions,” Vestnik FGUP NPO im. S. A. Lavochkina, No. 1, 53–62 (2009).

  55. O. V. Makarov, “On motion of a point of variable mass with minimal energy consumption,” Mechanics of Solids, No. 5, 178–173 (1977).

  56. M. B. Martynov, L. M. Zelyonyi, and V. V. Khartov, “Space program of research of planets and small bodies of the Solar system: principle of formation, concept of technical implementation,” Polet, No. 4, 107–118 (2011).

  57. L. I. Machabeli, “History of development of mechanics of bodies of variable mass from 1917 to 1967,” Vopr. Dinam. Prochn., No. 18, 181–194 (1969).

  58. I. V. Meshcherskii, Works on the Mechanics of a Body of Variable Mass [in Russian], Gostekhizdat, Moscow (1949).

    Google Scholar 

  59. G. K. Miikhailov, History of Dynamics of Systems of Variable Composition and Theory of Reactive Motion (before the Beginning of the Second World War) [in Russian], Preprint No. 49, Inst. Probl. Mekh..AN SSSR, Moscow (1974).

  60. N. N. Moiseev, Man, Environment, Society [in Russian], Nauka, Noscow (1982).

    Google Scholar 

  61. V. M. Kudryavtsev (ed.), Fundamentals of Theory and Computation of Liquid Rocket Engines [in Russian], Vyssh. Shkola, Moscow (1975).

    Google Scholar 

  62. D. E. Okhotsimskiy, “To the theory of rockets motion,” J. Appl. Math. Mech., 10, No. 2, 251–272 (1946).

    Google Scholar 

  63. A. S. Koroteev (ed.), Manned Expedition to Mars [in Russian], Rossiisk. Akad. Kosmon. im. K.E. Tsiolkovskogo, Moscow (2006).

    Google Scholar 

  64. L. I. Rozenoer, “The L. S. Pontryagin principle of maximum in the theory of optimal systems, II,” Automation and Remote Control, 20, No. 11, 1442–1458 (1959).

    Google Scholar 

  65. O. Ruban, One Stage is Sufficient, // n-t.ru/tp/ts/dos.htm

  66. L. V. Rykhlova, B. M. Shustov, V. G. Pol’, and K. G. Sukhanov, “Vital problems of the asteroid danger,” Near-Earth Astronomy, 25–33 (2007).

  67. F. P. Sanin and V. S. Savchuk, “On investigation of the history of development of the rocket space technique and physical-engineering problematics in Ukraine,” Vestn. Dnepropetr. Nats. Univ., Raketno-Kosm. Tekhn., No. 5, 11–16 (2001).

  68. V. A. Sapa, “On variational principles in mechanics of variable mass,” Izv. AN Kazakh. SSR, Ser. Mat. Mekh., No. 9, 116–123 (1961).

  69. T. Sato, K. Niita, V. A. Shurshakov, et al., “Estimation of a dose reduction in a spacecraft module on the use of water as supplementary protection,” Cosmic Research, 49, No. 4, 329–334 (2011).

    Article  ADS  Google Scholar 

  70. E. V. Tarasov, Cosmonautics [in Russian], Mashinostroenie, Moscow (1977).

    Google Scholar 

  71. Yu. M. Fatkin, “Inert mass use in engine of finite exhaust velocity,” Mechanics of Solids, No. 3, 164–168 (1967).

  72. R. P. Fedorenko, Approximate Solution of Optimal Control Problems [in Russian], Nauka, Moscow (1978).

    MATH  Google Scholar 

  73. E. A. Khasenov, Approximation of the Optimal Control in Degenerate Problems of Flight Mechanics [in Russian], Ph.D Thesis (Phys&Math), Kiev (1985).

  74. M. Hendeksen, “Optimal flight trajectory in the vacuum with constant thrust on the use of impulse trajectories as the initial approximations,” Raketn. Tekhn. Kosmonavt., No. 6, 151–158 (1966).

  75. P. G. Khorolskii, “Ballistic expedience of deep flexible control of the cruise engines of rockets-launchers,” Aviats.-Kosm. Tekhn. Tekhnol., No. 10, 11–13 (2006).

  76. A. S. Shakhverdyan and S. V. Shakhverdyan, “Theory of singular optimal controls with application to mechanics of space flight,” Cosmic Research, 42, No. 3, 302–312 (2004).

    Article  MATH  Google Scholar 

  77. G. P. Shibanov, Inhabitation of Space and the Safety of Man in It [in Russian], Mashinostroenie, Moscow (2007).

    Google Scholar 

  78. A. G. Accettura, C. Bruno, S. Casotto, and F. Marzari, “Mission to Mars using integrated propulsion concepts: considerations, opportunities, and strategies,” Acta Astronautica, 54, No. 7. 471–486 (2004).

    Article  ADS  Google Scholar 

  79. M. S. Aronna, J. F. Bonnans, and P Martinon, “A shooting algorithm for optimal control problems with singular arcs,” J. Optimiz. Theor. Appl., 158, No. 2, 419–459 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  80. R. Beichel, C. J. O’Brien, and J. P. Taylor, “The next generation rocket engines,” Acta Astronautica, 20, 111–116 (1989).

    Article  ADS  Google Scholar 

  81. J. T. Betts, “Survey of numerical methods for trajectory optimization,” J. Guid. Contr. Dynam., 21, No. 2, 193–207 (1998).

    Article  ADS  MATH  Google Scholar 

  82. P. Caisso, A. Souchier, C. Rothmund, et al., “A liquid propulsion panorama,” Acta Astronautica, 65, No. 11, 1723–1737 (2009).

  83. V. P. Chiravalle, “Nuclear electric ion propulsion for three deep space missions,” Acta Astronautica, 62, No. 6, 374–390 (2008).

    Article  ADS  Google Scholar 

  84. A. S. Dhoble and P. C. Pullammanappallil, “Design and operation of an anaerobic digester for waste management and fuel generation during long term lunar mission,” Advances in Space Research, 54, No. 8, 1502–1512 (2014).

    Article  ADS  Google Scholar 

  85. B. L. Ehlmann, J. Chowdhury, T.C. Marzullo, et al., “Humans to Mars: A feasibility and cost–benefit analysis,” Acta Astronautica, 56, No. 9, 851–858 (2005).

    Article  ADS  Google Scholar 

  86. G. G. Fedotov, “Possibilities of combining high- and low-thrust engines in flights to Mars,” Acta Astronautica, 55, 79–94 (2004).

    Article  Google Scholar 

  87. F. W. Gobetz and J. R. Doll, “A survey of impulsivå trajectories,” AIAA J., 7, No. 5, 801–834 (1969).

    Article  ADS  Google Scholar 

  88. R. H. Goddard, “A method of reaching extreme altitudes,” Nature, 105, No. 2, 809–811 (1920).

    Article  ADS  Google Scholar 

  89. A. N. Guz, V. D. Kubenko, and A. E. Babaev, “Dynamics of shell systems interacting with a liquid,” Int. Appl. Mech., 38, No. 3, 260–301 (2002).

    Article  ADS  MATH  Google Scholar 

  90. G. Hamel, “Uber eine mit dem Problem der Rakete zusammenhangende Aufgabe der Variationsrechnung,” ZAMM, 7, No. 6, 451–452 (1927).

    MATH  Google Scholar 

  91. S. D. Howe, “High energy-density propulsion—reducing the risk to humans in planetary exploration,” Space Policy, No. 17, 275–283 (2001).

  92. A. V. Ilin, L. D. Cassidy, T. W. Glover, and F. R. Chang Díaz, “VASIMR ®Human Mission to Mars,” Space, Propulsion & Energy Sciences Int. Forum (March 15 – 17, University of Maryland), 12 p. (2011).

  93. V. V. Ivashkin, “Analysis of space flight mechanics problems,” Acta Astronautica, 52, No. 8, 663–670 (2003).

    Article  ADS  Google Scholar 

  94. H. J. Kelley, R. E. Kopp, and H. G. Moyer, “Singular extremals,” in: G. Leitmann (ed.), Topics in Optimization, Acad. Press, N.Y. (1967), pp. 63–101.

    Chapter  Google Scholar 

  95. O. M. Kharitonov and B. M. Kiforenko, “Finite-thrust optimization of interplanetary transfers of space vehicle with bimodal nuclear thermal propulsion,” Acta Astronautica, 69, No. 3, 223–233 (2011).

    Article  ADS  Google Scholar 

  96. B. N. Kiforenko, “Problems of the mathematical description of rocket engines as plants,” Int. Appl. Mech., 48, No. 5, 608–612 (2012).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  97. B. N. Kiforenko and I. Yu. Vasil’ev, “How we will go to Mars,” Acta Astronautica, 54, No. 1, 61–67 (2003).

    Article  ADS  Google Scholar 

  98. B. N. Kiforenko and I. Yu. Vasil’ev, “How Shall We Do Go to Mars,” Proc. 46th Int. Astronautical Cong. (IAC-95), (Oslo, Norway, 1995), IAF Pap. 95–A.6.0, 9 p. (1995).

  99. B. N. Kiforenko, I. Yu. Vasil’ev, and Ya. V. Tkachenko, “On the problem of optimal control of the thrust value of the electric propulsion rocket with solar energy source,” Acta Astronautica, 89, August–September, 121–125 (2013).

  100. H. H. Koelle and D. G. Stephenson, “International Academy of Astronautics 5th cosmic study – preparing for a 21st century program of integrated, Lunar and Martian exploration and development (executive summary), Acta Astronautica, 52, No. 8, 649–662 (2003).

    Article  ADS  Google Scholar 

  101. M. S. Konstantinov and V. G. Petukhov, “The analysis of manned Mars mission with duration of 1000 days,” Acta Astronautica, 73, 122–136 (2012).

    Article  ADS  Google Scholar 

  102. R. Kumar and H. J. Kelley, “Singular optimal atmospheric rocket trajectories," J. Guid. Contr. Dynam., 11, No. 4, 305–312 (1988).

    Article  ADS  MATH  Google Scholar 

  103. D. F. Landau and, J. M. Longuski, “Comparative assessment of human-Mars-mission technologies and architectures,” Acta Astronautica, 65, No. 7, 893–911 (2009).

  104. V. B. Larin, “Attitude-determination problems for rigid body,” Int. Appl. Mech., 37, No. 7, 870–898 (2001).

    Article  ADS  MathSciNet  Google Scholar 

  105. V. B. Larin, “Compensation of external perturbations under uncertainty,” Int. Appl. Mech., 38, No. 9, 1145–1150 (2002).

    Article  ADS  MATH  Google Scholar 

  106. V. B. Larin and A. A. Tunik, “On inertial-navigation system without angular-rate sensors,” Int. Appl. Mech., 49, No. 4, 488–499 (2013).

    Article  ADS  Google Scholar 

  107. D. F. Lawden, “Optimal intermediate-thrust arcs in a gravitational field,” Acta Astronautica, 8, No. 2, 106–123 (1962).

    Google Scholar 

  108. C. Marshal and P. Contensou “Singularities in optimization of deterministic dynamic systems,” J. Guid. Contr. Dynam., 4, No. 3, 240–252 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  109. P. Martinon, F. Bonnans, J. Laurent-Varin, and E. Trélat, “Numerical study of optimal trajectories with singular arcs for an Ariane 5 launcher,” J. Guid. Contr. Dynam., 32, No. 1, 51–55 (2009).

    Article  ADS  Google Scholar 

  110. A.A. Martynyuk, “Stability analysis of continuous systems with structural perturbations,” Int. Appl. Mech., 38, No. 7, 783–805 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  111. D. D. Mazanek, J. R. Brohpy, and R. G. Merrill, “Asteroid retrieval mission concept–trailblazing our future in space and helping to protect us from Earth impactors,” in: Proc 3rd IAA Planetary Defense Conf., April 15 – 19. 2013; Flagstaff, AZ; United States, 16 p. (2013).

  112. T. T. McConaghy, J. L. Longuski, and D. V. Byrnes, “Analysis of a class of Earth–Mars cycler trajectories,” J. Spacecraft and Rockets, 41, No. 4, 622–628 (2004).

    Article  ADS  Google Scholar 

  113. R. X. Meyer, “The “Flying Carpet” concept: A possible alternative to nuclear space propulsion,” Acta Astronautica, 58, No. 10, 499–505 (2006).

    Article  ADS  Google Scholar 

  114. A. Miele, “General variational theory of the flight paths of rocket-powered aircraft, missile, and satellite carriers,” Acta Astronautica, 4, No. 4, 264–288 (1958).

    Google Scholar 

  115. H. Munick, “Goddard problem with bounded thrust,” AIAA J., 3, No. 7, 1283–1285 (1965).

    Article  ADS  MathSciNet  Google Scholar 

  116. H. J. Oberle, “Numerical computation of singular control functions in trajectory optimization problems,” J. Guid. Contr. Dynam., 13, No. 1, 153–159 (1990).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  117. F. Renk, M. Hechler, and E. Messerschmid, “Exploration missions in the Sun–Earth–Moon system: A detailed view on selected transfer problems,” Acta Astronautica, 67, No. 1, 82–96 (2010).

    Article  ADS  Google Scholar 

  118. R. J. Salkeld and R. Beichel, “Mixed–mode rocket vehicles for international space transportation systems,” Acta Astronautica, 4, No. 1, 213–227 (1977).

    Article  ADS  Google Scholar 

  119. S. N. Sanoob, M. G. Prince, and B. Sundar, “Numerical analysis of aero-spike nozzle for spike length optimization,” Int. J. Res. Eng. Technol., 1, No. 6, 1–14 (2013).

    Google Scholar 

  120. H. Seywald and E. M. Cliff, “Goddard problem in presence of a dynamic pressure limit,” J. Guid. Contr. Dynam., 16, No. 4, 776–781 (1993).

    Article  ADS  MATH  Google Scholar 

  121. R. D. Shaver, “The two-stage sounding rocket problem,” J. Spacecraft and Rockets, 4, No. 10, 1310–1315 (1967).

    Article  ADS  Google Scholar 

  122. Y. Sinyak, A. Grigoriev, V. Gaydadimov, et al., “Deuterium-free water (1H2O) in complex life-support systems of long-term space missions,” Acta Astronautica, 52, No. 7, 575–580 (2003).

    Article  ADS  Google Scholar 

  123. N. Strange, T. M. Landau, G. Lantoine, et al., “Overview of mission design for NASA asteroid redirect robotic mission concept,” in: Proc. 33rd Int. Electric Propulsion Conf., The George Washington University, Washington (2013), p. 13.

  124. L. Summerer, “Thinking tomorrows’ space – Research trends of the ESA advanced concepts team 2002–2012,” Acta Astronautica, 95, February–March, 242–259 (2014).

  125. V. B. Òawakley, “On optimization problem of rocket vehicles for a generalized thrust characteristics,” Zeitschrift für Flugwissenschaften, 17, No. 9, 305–312 (1969).

    Google Scholar 

  126. M. Taraba, K. Zwintz, C. Bombardelli, et al., “Project M3 – a study for a manned Mars mission in 2031,” Acta Astronautica, 58, No. 2, 88–104 (2006).

    Article  ADS  Google Scholar 

  127. The Space Review: VASIMR: hope or hype for Mars exploration? http://t.co/YyHg9xAPns.

  128. E. Tre’lat, “Optimal control and applications to aerospace: some results and challenges,” J. Optimiz. Theor. Appl., 154, No. 3, 713–758 (2012).

    Article  MathSciNet  Google Scholar 

  129. P. Tsiotras and H. J. Kelley, “Drag-law effects in the Goddard problem,” Automatica, 27, No. 3, 481–490 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  130. P. Tsiotras and H. J. Kelley, “Goddard problem with constrained time of flight,” J. Guid. Contr. Dynam., 15, No. 2, 289–296 (1992).

    Article  ADS  MATH  Google Scholar 

  131. E. E. Weeks and A. A. Faiyetole, “Science, technology and imaginable social and behavioral impacts as outer space develops,” Acta Astronautica, 95, 166–173 (2014).

    Article  ADS  Google Scholar 

  132. J. C. Willems, A. Kitapci, and L. M. Silverman, “Singular optimal control: a geometric approach,” SIAM J. Contr. Optimiz., 24, No. 2, 323–337 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  133. F. H. Winter, M. J. Neufeld, and K. Dougherty “Was the rocket invented or accidentally discovered? Some new observations on its origins," Acta Astronautica, 77, 131–137 (2012).

    Article  ADS  Google Scholar 

  134. A. E. Zakrzhevskii, “The dynamics of systems of rigid and elastic bodies as applied to spacecraft,” Int. Appl. Mech., 36, No. 8, 1001–1036 (2000).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to B. N. Kiforenko.

Additional information

Translated from Prikladnaya Mekhanika, Vol. 53, No. 3, pp. 3–62, May–June, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kiforenko, B.N. Singular Optimal Controls of Rocket Motion (Survey). Int Appl Mech 53, 237–286 (2017). https://doi.org/10.1007/s10778-017-0809-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10778-017-0809-3

Keywords

Navigation