Optimal Maneuvers with Bounded Inputs

  • Ashish Tewari
Part of the Control Engineering book series (CONTRENGIN)


Spacecraft engines can generate only limited thrust magnitudes. This implies that the acceleration inputs in space navigation must necessarily be bounded. The trajectory optimization must therefore be performed taking bounded inputs into account. As discussed in the earlier chapters, the nature of optimal trajectories with bounded acceleration inputs can be classified as either impulsive thrust or continuous thrust maneuvers.


  1. 3.
    Archenti, A.R., Vinh, N.X.: Intermediate thrust arcs and their optimality in a central force field. J. Optim. Theory Appl. 11, 293–304 (1973)MathSciNetCrossRefGoogle Scholar
  2. 4.
    Athans, M., Falb, P.L.: Optimal Control. Dover, New York (2007)zbMATHGoogle Scholar
  3. 8.
    Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series, Reston (1999)Google Scholar
  4. 9.
    Bell, D.J.: Optimal space trajectories – a review of published work. Aeronaut. J. R. Aeronaut. Soc. 72, 141–146 (1968)Google Scholar
  5. 10.
    Bell, D.J.: Second variation and singular space trajectories. Int. J. Control 14, 697–703 (1971)CrossRefGoogle Scholar
  6. 11.
    Bell, D.J.: The non-optimality of Lawden’s spiral. Acta Astronaut. 16, 317–324 (1971)MathSciNetzbMATHGoogle Scholar
  7. 12.
    Bell, D.J., Jacobson, D.H.: Singular Optimal Control Problems. Academic, New York (1975)zbMATHGoogle Scholar
  8. 16.
    Bolonkin, A.A.: Special Extrema in Optimal Control Problems. Eng. Cybern. 2, 170–183 (1969)MathSciNetGoogle Scholar
  9. 17.
    Breakwell, J.V.: A doubly singular problem in optimal interplanetary guidance. SIAM J. Control 3, 71–77 (1965)MathSciNetGoogle Scholar
  10. 19.
    Bryson, A.E., Ho, Y.: Applied Optimal Control. Wiley, New York (1979)Google Scholar
  11. 21.
    Carter, T.: Fuel-optimal maneuvers of a spacecraft relative to a point in circular orbit. J. Guid. Control Dyn. 7, 710–716 (1984)CrossRefGoogle Scholar
  12. 26.
    Da Silva, F.S.: Solution of the coast-arc problem using Sundman transformation. Acta Astronaut. 50, 1–11 (2001)Google Scholar
  13. 31.
    Forbes, G.F.: The trajectory of a powered rocket in space. J. Br. Interplanet. Soc. 9, 75–79 (1950)Google Scholar
  14. 32.
    Gabasov, R.: Necessary conditions for optimality of singular control. Eng. Cybern. 5, 28–37 (1968)MathSciNetGoogle Scholar
  15. 34.
    Goh, B.S.: Compact forms of the generalized Legendre conditions and the derivation of the singular extremals. In: Proceedings of 6th Hawaii International Conference on System Sciences, pp. 115–117 (1973)Google Scholar
  16. 39.
    Hermes, H.: Controllability and the singular problem. SIAM J. Control 2, 241–260 (1964)MathSciNetzbMATHGoogle Scholar
  17. 42.
    Jacobson, D.H.: A new necessary condition of optimality for singular control problems. SIAM J. Control 7, 578–595 (1969)MathSciNetCrossRefGoogle Scholar
  18. 45.
    Kelley, H.J.: Singular extremals in Lawden’s problem of optimal rocket flight. AIAA J. 1, 1578–1580 (1963)CrossRefGoogle Scholar
  19. 47.
    Kopp, R.E., Moyer, H.G.: Necessary conditions for singular extremals. AIAA J. 3, 1439–1444 (1965)CrossRefGoogle Scholar
  20. 49.
    Lawden, D.F.: Inter-orbital transfer of a rocket. Annual Report, pp. 321–333. British Interplanetary Society (1952)Google Scholar
  21. 50.
    Lawden, D.F.: Fundamentals of space navigation. J. Br. Interplanet. Soc. 13, 87–101 (1954)Google Scholar
  22. 51.
    Lawden, D.F.: Optimal intermediate-thrust arcs in a gravitational field. Acta Astronaut. 8, 106–123 (1954)Google Scholar
  23. 52.
    Lawden, D.F.: Optimal Trajectories for Space Navigation. Butterworths, London (1963)zbMATHGoogle Scholar
  24. 53.
    Leitmann, G.: On a class of variational problems in rocket flight. J. Aerospace Sci. 26, 586–591 (1959)MathSciNetCrossRefGoogle Scholar
  25. 57.
    Marec, J.P.: Optimal Space Trajectories. Elsevier, New York (1979)zbMATHGoogle Scholar
  26. 61.
    Minter, C.F., Fuller-Rowell, T.J.: A robust algorithm for solving unconstrained two-point boundary value problems. In: Proceedings of AAS/AIAA Spaceflight Mechanics Meeting, AAS Paper 05-129, Copper Mountain (2005)Google Scholar
  27. 65.
    Pan, B., Zheng, C., Lu, P., Gao, B.: Reduced transversality conditions in optimal space trajectories. J. Guid. Control. Dyn. 36, 1289–1300 (2013)CrossRefGoogle Scholar
  28. 67.
    Pines, S.: Constants of the motion for optimum thrust trajectories in a central force field. AIAA J. 2, 2010–2014 (1964)MathSciNetCrossRefGoogle Scholar
  29. 71.
    Robbins, H.M.: Optimality of intermediate-thrust arcs of rocket trajectories. AIAA J. 3, 1094–1098 (1965)MathSciNetCrossRefGoogle Scholar
  30. 75.
    Snow, D.R.: Singular optimal controls for a class of minimum effort problems. SIAM J. Control 2, 203–219 (1964)MathSciNetzbMATHGoogle Scholar
  31. 81.
    Tewari, A.: Advanced Control of Aircraft, Spacecraft, and Rockets. Wiley, Chichester (2011)CrossRefGoogle Scholar
  32. 83.
    Fraeijs de Veubeke, B.: Canonical transformations and the thrust-coast-thrust optimal transfer problem. Acta Astronaut. 11, 271–282 (1965)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ashish Tewari
    • 1
  1. 1.Department of Aerospace EngineeringIndian Institute of Technology, KanpurIIT-KanpurIndia

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