Advertisement

Relaxation Gaps in Optimal Control Processes with State Constraints

  • Sandra Butzek
  • Werner H. Schmidt
Part of the International Series of Numerical Mathematics book series (ISNM, volume 124)

Abstract

There are a lot of very simple examples of control processes which have no optimal solution. A well-known fundamental existence theorem is that of Roxin-Fillipov; unfortunately, the assumptions in this theorem are rather strong. Sometimes one can prove existence in absence of convexity by studying “bigger” problems obtained by relaxation, that means by convexification of the sets of speed vectors. Then we try to choose a special optimal solution of the relaxed problem and apply PONTRYAGIN’S maximum principle in order to discuss whether certain derived controls are optimal ones of the original problem (or not). BALDER [1] used a similar idea to prove new existence results for optimal control problems without convexity. He applies BAUER’S extremal principle instead of the maximum principle.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Balder, E.J.: New existence results for optimal controls in the absence of convexity: The importance of extremality. SIAM J. Control and Optimization, Vol. 33 (1994), No. 3, pp. 890–916.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Bauer, H.: Minimalstellen von Funktionen und Extremalpunkte. Archiv d. Math. 9 (1958), 389–393, 11 (1960)), 200-205.zbMATHCrossRefGoogle Scholar
  3. [3]
    Bittner, L.: Zur Konvexifizierung von Optimalsteuerproblemen. Report SFB 255, Nr.33, Greifswald und München, 1996.Google Scholar
  4. [4]
    Butzek, S.; Schmidt, W.H.: Konstruktion von Näherungslösungen für Steuerprobleme mit Hilfe von Lösungen relaxierter Probleme. Report SFB 255, Nr. 26, Greifswald und München, 1996.Google Scholar
  5. [5]
    Butzek, S.: Aspekte des Zusammenhangs von Optimalsteuerproblemen und zugehörigen relaxierten Problemen. PhD-thesis. Greifswald 1997.Google Scholar
  6. [6]
    Roubicek, T.: Relaxation in Optimization Theory and Variational Calculus. W.De Gruyter-Verlag. Berlin 1997zbMATHCrossRefGoogle Scholar
  7. [7]
    Schmidt, W.H.: An existence theorem for a special control problem. Proceedings of MMAR′96. Vol. I, 263-266. Szczecin, 1996.Google Scholar
  8. [8]
    Tikhomirov, V.M.: Grundprinzipien der Theorie der Extremalaufgaben. Teubner-Texte zur Mathematik. Bd. 30. Leipzig, 1982.Google Scholar
  9. [9]
    Warga, J.: Relaxed Variational Problems. Journal of Math. Analysis and Appl. 4, 111–128, 1962.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1998

Authors and Affiliations

  • Sandra Butzek
    • 1
  • Werner H. Schmidt
    • 1
  1. 1.Institut für Mathematik und InformatikErnst-Moritz-Arndt-Universität GreifswaldGreifswaldGermany

Personalised recommendations