Advertisement

Numerical and Analytical Methods for Global Optimization

  • Paolo Teofilatto
  • Mauro Pontani
Conference paper
Part of the Springer Optimization and Its Applications book series (SOIA, volume 33)

Abstract

Global optimization is an important issue in the field of optimal aerospace trajectories. The joint use of global (approximate) numerical methods and of accurate local methods is one of the current approaches to global optimization. Nevertheless, the existence of global analysis techniques for investigating the possible multiplicity of results in optimization problems is of great interest. One of these techniques was the analysis of optimal trajectories through the Green’s theorem. This approach, formerly introduced by Miele, has been applied to find singular optimal solutions related to missile, spacecraft, and aircraft trajectories. Another approach is based on the Morse theory, which relates the number of singular points of the objective function to the topology of the space where this function is defined. In particular, the so-called Morse inequalities provide a lower bound on the number of the local minima of the objective function. In this paper, Miele’s and Morse’s approaches will be recalled and applied to some problems in flight mechanics.

Keywords

Cost Function Global Optimization Optimal Control Problem Global Optimal Solution Morse Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Teofilatto, P., De Pasquale, E.: A non linear adaptive guidance for last stage control. Journal of Aerospace Engineering, Part G 213, 45–55 (1999)CrossRefGoogle Scholar
  2. 2.
    Miele, A.: Minimum time in nonsteady flight of aircraft (in Italian). Atti della Accademia delle Scienze di Torino, 85, 41–52 (1950)Google Scholar
  3. 3.
    Miele, A.: General optimal solutions for aircraft in nonsteady flight (in italian). L’Aerotecnica, 32, 135–142 (1952)Google Scholar
  4. 4.
    Miele, A.: Optimal flight trajectories of turbojet aircraft (in Italian). L’Aerotecnica, 32, 206–219 (1952)Google Scholar
  5. 5.
    Miele, A.: On non steady climb of turbojet aircraft. Journal of the Astronautical Sciences, 21, 781–783 (1954)MATHGoogle Scholar
  6. 6.
    Miele, A.: Minimum time flight trajectories (in Italian). Atti della Accademia delle Scienze di Torino, 21, 80–87 (1954)Google Scholar
  7. 7.
    Cicala, P., Miele, A.: Brachistocronic maneuvers of constant mass, Journal of the Astronautical Sciences, 2, 286–288 (1955)Google Scholar
  8. 8.
    Miele, A.: General variational theory of flight paths of rocket powered aircraft, missiles and satellite carriers, Astronautica Acta, 4, 11–21 (1958)Google Scholar
  9. 9.
    Miele, A.: Flight mechanics and variational problems of linear type, Journal of the Astronautical Sciences, 25, 286–288 (1958)Google Scholar
  10. 10.
    Miele, A.: Extremization of linear integrals by Green’s theory. In: Leitmann, G. (ed) Optimization Techniques. Academic Press (1962)Google Scholar
  11. 11.
    YamarË, T. et al.: Start up of chemostat: application of fed-batch culture, Biotechnology and Bioengineering, 21, 111–129 (1978)Google Scholar
  12. 12.
    Weigard, W.: Maximum cell productivity by repeated fed-batch culture, Biotechnology and Bioengineering, 23, 249–266 (1980)CrossRefGoogle Scholar
  13. 13.
    Constantiniedes, A.: Application of optimization methods to the control of fermentation processes, Annals of the New York Academy of Sciences, 326, 193–221 (1979)CrossRefGoogle Scholar
  14. 14.
    Sethi, S.: Optimal institutional advertising: minimum time problems, Journal of Optimization theory and applications, 14, 213–231 (1974)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Sethi, S.: Optimal advertising policy with the contageon model, Journal of Optimization theory and applications, 29, 615–627 (1979)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sethi, S.: Optimal quarantine programs for controlling epidemic spread, Journal of Optimization theory and applications, 29, 202–212 (1978)Google Scholar
  17. 17.
    Stabble, P., Maronsky, R.: Minimum time running and swimming, Journal of Biomechanics, 29, 245–249 (1996)CrossRefGoogle Scholar
  18. 18.
    Milnor, J.: Morse theory. Princeton University Press, Princeton (1969)Google Scholar
  19. 19.
    Edelbaum, P.: How many impulses ? , Astronautics and Aeronautics, 5, 245–249 (1997)Google Scholar
  20. 20.
    Hazelrigg, G.: Globally optimal impulsive transfers via Green’s theorem, Journal of Guidance, Control, and Dynamics, 7, 462–470 (1984)MATHCrossRefGoogle Scholar
  21. 21.
    Lawden, D.: Optimal trajectories for space navigation. Butterworths, London (1963)Google Scholar
  22. 22.
    Pontani, M.: Simple Method to Determine Globally Optimal Orbital Transfers. Journal of Guidance, Control and Dynamics, 32, No.3, 899–914 (2009)Google Scholar
  23. 23.
    Hermes, H., Heynes, G.: On nonlinear control problem with control appearing linearly, SIAM Journal of Control, 1, 85–108 (1963)MATHGoogle Scholar
  24. 24.
    Heynes, G.: The optimality of a totally singular vector control: an extension of the Green’s theorem to higher dimensions, SIAM Journal of Control, 4, 662–677 (1966)CrossRefGoogle Scholar
  25. 25.
    Neustadt, L.: Minimum effort control systems, SIAM Journal of Control, 1, 16–31 (1962)MATHMathSciNetGoogle Scholar
  26. 26.
    Spainer, E.: Algebraic topology, Mc Graw-Hill, New York (1966)Google Scholar
  27. 27.
    Palis, R., Smale, S.: A generalized Morse theory, Bulletin of the American Mathematical Society, 70, 165–172 (1964)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Agrachev, A.A., Vakhrammev, S.A.: Morse theory and optimal control problems. In Progress in System Control Theory, Birkhauser, Boston 1–11 (1991)Google Scholar
  29. 29.
    Vakhrammev, S.A.: Hilbert manifolds with corners of finite codimension and the theory of optimal control, Journal of Mathematical Sciences, 53, 176–223 (1991)CrossRefGoogle Scholar
  30. 30.
    Vakhrammev, S.A.: Morse theory and the Lyusternik-Shnirelman theory in geometric control theory, Journal of Mathematical Sciences, 71, 2434–2485 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Scuola di Ingegneria AerospazialeUniversity of Rome “La Sapienza,”Italy
  2. 2.Scuola di Ingegneria AerospazialeUniversity of Rome “La Sapienza,”Italy

Personalised recommendations