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Journal of Optimization Theory and Applications

, Volume 18, Issue 2, pp 235–257 | Cite as

Numerical solution of singular control problems using multiple shooting techniques

  • H. Maurer
Contributed Papers

Abstract

Algorithms for calculating the junction points between optimal nonsingular and singular subarcs of singular control problems are developed. The algorithms consist in formulating appropriate initialvalue and boundary-value problems; the boundary-value problems are solved with the method of multiple shooting. Two examples are detailed to illustrate the proposed numerical methods.

Key Words

Control theory singular problems numerical methods aerospace engineering economic planning 

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References

  1. 1.
    Gabasov, R., andKirillova, F. M.,High Order Necessary Conditions for Optimality, SIAM Journal on Control, Vol. 10, No. 1, 1972.Google Scholar
  2. 2.
    Goh, B. S.,The Second Variation for the Singular Bolza Problem, SIAM Journal on Control, Vol. 4, No. 2, 1966.Google Scholar
  3. 3.
    Goh, B. S.,Necessary Conditions for Singular Extremals Involving Multiple Control Variables, SIAM Journal on Control, Vol. 4, No. 4, 1966.Google Scholar
  4. 4.
    Jacobson, D. H.,A New Necessary Condition of Optimality for Singular Control Problems, SIAM Journal on Control, Vol. 7, No. 4, 1969.Google Scholar
  5. 5.
    Kelley, H. J., Kopp, R. E., andMoyer, H. G.,Singular Extremals, Topics in Optimization, Edited by G. Leitmann, Academic Press, New York, New York, 1967.Google Scholar
  6. 6.
    McDanell, J. P., andPowers, W. F.,Necessary Conditions for Joining Optimal Singular and Nonsingular Subarcs, SIAM Journal on Control, Vol. 9, No. 2, 1971.Google Scholar
  7. 7.
    Jacobson, D. H.,Sufficient Conditions for Nonnegativity of the Second Variation in Singular and Nonsingular Control Problems, SIAM Journal on Control, Vol. 8, No. 3, 1970.Google Scholar
  8. 8.
    McDanell, J. P., andPowers, W. F.,New Jacobi-Type Necessary and Sufficient Conditions for Singular Optimization Problems, AIAA Journal, Vol. 8, pp. 1416–1420, 1970.Google Scholar
  9. 9.
    Speyer, J. L., andJacobson, D. H.,Necessary and Sufficient Conditions for Optimality for Singular Control Problems: A Transformation Approach, Journal of Mathematical Analysis and Applications, Vol. 33, No. 1, 1971.Google Scholar
  10. 10.
    Jacobson, D. H., Gershwin, S. B., andLele, M. M.,Computation of Optimal Singular Controls, IEEE Transactions on Automatic Control, Vol. 15, 1970.Google Scholar
  11. 11.
    Pagurek, B., andWoodside, C. M.,The Conjugate Gradient Method for Optimal Control Problems with Bounded Control Variables, Automatica, Vol. 4, 1968.Google Scholar
  12. 12.
    Soliman, M. A., andRay, W. H.,A Computational Technique for Optimal Control Problems Having Singular Arcs, International Journal on Control, Vol. 16, No. 2, 1972.Google Scholar
  13. 13.
    Bulirsch, R.,Die Mehrzielmethode zur Numerischen Lösung von Nichtlinearen Randwertproblemen und Aufgaben der Optimalen Steuerung, Vortrag im Lehrgang “Flugbahnoptimierung” der Carl-Cranz-Gesellschaft, West Germany, 1971.Google Scholar
  14. 14.
    Bulirsch, R., Stoer, J., andDeuflhard, P.,Numerical Solution of Nonlinear Two-Point Boundary Value Problems, I, Handbook Series Approximation, Numerische Mathematik (to appear).Google Scholar
  15. 15.
    Keller, H. B.,Numerical Methods for Two-Point Boundary Value Problems, Blaisdell Publishing Company, Waltham, Massachusetts, 1968.Google Scholar
  16. 16.
    Hermes, H.,Controllability and the Singular Problem, SIAM Journal on Control, Vol. 2, No. 2, 1964.Google Scholar
  17. 17.
    Johnson, C. D.,Singular Solutions in Optimal Control Problems, Advances in Control Systems, Vol. 2, Edited by C. T. Leondes, Academic Press, New York, New York, 1965.Google Scholar
  18. 18.
    Maurer, H.,Zur Numerischen Berechnung Optimaler Singulärer Steuerungen, Mathematisches Institut der Universität Köln, Ph.D. Thesis, 1972.Google Scholar
  19. 19.
    Dickmanns, E. D.,The Effect of Finite Thrust and Heating Constraints on the Synenergetic Plane Change Maneuver for a Space Shuttle Orbiter Class Vehicle, NASA, TN No. 7311.Google Scholar
  20. 20.
    Zimmermann, U.,Numerische Berechnung Optimaler Steuerungen mit Mehrfacher Beschränkung im Phasenraum bei Problemen der Flugbahnoptimierung durch Anwendung der Mehrzielmethode, Mathematisches Institut der Universität Köln, Diploma Thesis, 1973.Google Scholar
  21. 21.
    Bryson, A. E., andHo, Y. C.,Applied Optimal Control, Blaisdell Publishing Company, Waltham, Massachusetts, 1969.Google Scholar
  22. 22.
    Dobell, A. R., andHo, Y. C.,Optimal Investment Policy, Mathematical Systems Theory and Economics, Vol. 1, Edited by H. Kuhn and G. Szego, Springer Verlag, Berlin, Germany, 1969.Google Scholar
  23. 23.
    Garfinkel, B.,A Solution of the Goddard Problem, SIAM Journal on Control, Vol. 1, No. 3, 1963.Google Scholar
  24. 24.
    Halkin, H.,Method of Convex Ascent, Computing Methods in Optimization Problems, Edited by A. V. Balakrishnan and L. W. Neustadt, Academic Press, New York, New York, 1964.Google Scholar
  25. 25.
    Miele, A.,Extremization of Linear Integrals by Green's Theorem, Optimization Techniques, Edited by G. Leitmann, Academic Press, New York, New York, 1962.Google Scholar
  26. 26.
    Regenberg, S.,Teil II: Numerische Behandlung von Optimierungsproblemen, Optimierungsverfahren, Edited by H. Tolle and S. Regenberg, Brennpunkt für Navigation, Technische Universität Berlin, 1966.Google Scholar
  27. 27.
    Tsien, H. S., andEvans, R. C.,Optimal Thrust Programming for a Sounding Rocket, Journal of the American Rocket Society, Vol. 21, No. 5, 1951.Google Scholar
  28. 28.
    Diekhoff, H. J.,Über die Optimale Steuerung einer Höhenrakete, Mathematisches Institut der Universität Köln, Diploma Thesis, 1973.Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • H. Maurer
    • 1
  1. 1.Mathematisches Institut der Universitat Würzburg, Am HublandWürzburgWest Germany

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