Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Maximum Principle in Optimal Control

  • Velimir Jurdjevic
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-1806-1_57

Article Outline

Glossary

Definition of the Subject

Introduction

The Calculus of Variations and the Maximum Principle

Variational Problems with Constraints

Maximum Principle on Manifolds

Abnormal Extrema and Singular Problems

Future Directions

Bibliography

Keywords

Manifold 
This is a preview of subscription content, log in to check access

Bibliography

Primary Literature

  1. 1.
    Agrachev AA, Sachkov YL (2005) Control theory from the geometric viewpoint. Encycl Math Sci 87. Springer, HeidelbergGoogle Scholar
  2. 2.
    Arnold VI (1989) Mathematical methods of classical mechanics. Graduate texts in Mathematics, vol 60. Springer, HeidelbergCrossRefGoogle Scholar
  3. 3.
    Bonnard B, Chyba M (2003) Singular trajectories and their role in control. Springer, HeidelbergMATHGoogle Scholar
  4. 4.
    Bianchini RM, Stefani G (1993) Controllability along a trajectory; a variational approach. SIAM J Control Optim 31:900–927MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bressan A (2007) On the intersection of a Clarke cone with a Boltyanski cone (to appear)Google Scholar
  6. 6.
    Berkovitz LD (1974) Optimal control theory. Springer, New YorkMATHCrossRefGoogle Scholar
  7. 7.
    Caratheodory C (1935) Calculus of variations. Teubner, Berlin (reprinted 1982, Chelsea, New York)Google Scholar
  8. 8.
    Clarke FH (1983) Optimization and nonsmooth analysis. Wiley Interscience, New YorkMATHGoogle Scholar
  9. 9.
    Clarke FH (2005) Necessary conditions in dynamic optimization. Mem Amer Math Soc 816(173)Google Scholar
  10. 10.
    Fomenko AT, Trofimov VV (1988) Integrable systems on Lie algebras and symmetric spaces. Gordon and BreachGoogle Scholar
  11. 11.
    Gamkrelidze RV (1978) Principles of optimal control theory. Plenum Press, New YorkMATHCrossRefGoogle Scholar
  12. 12.
    Jurdjevic V (1997) Geometric control theory. Cambridge Studies in Advanced Mathematics vol 51. Cambridge University Press, CambridgeGoogle Scholar
  13. 13.
    Jurdjevic V (2005) Hamiltonian systems on complex Lie groups and their homogeneous spaces. Mem Amer Math Soc 178(838)MathSciNetGoogle Scholar
  14. 14.
    Lee EB, Markus L (1967) Foundations of optimal control theory. Wiley, New YorkMATHGoogle Scholar
  15. 15.
    Liu WS, Sussmann HJ (1995) Shortest paths for SR metrics of rank 2 distributions. Mem Amer Math Soc 118(564)MathSciNetGoogle Scholar
  16. 16.
    Knobloch H (1975) High order necessary conditions in optimal control. Springer, BerlinGoogle Scholar
  17. 17.
    Krener AJ (1977) The high order maximum principle and its application to singular extremals. SIAM J Control Optim 17:256–293MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kupka IK (1990) The ubiquity of Fuller’s phenomenon. In: Sussmann HJ (ed) Non-linear controllability and Optimal control. Marcel Dekker, New York, pp 313–350Google Scholar
  19. 19.
    Marchal C (1973) Chattering arcs and chattering controls. J Optim Theory App 11:441–468MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Morduchovich B (2006) Variational analysis and generalized differentiation: I. Basic analysis, II. Applications. Grundlehren Series (Fundamental Principle of Mathematical Sciences). Springer, BerlinCrossRefGoogle Scholar
  21. 21.
    Montgomery R (1994) Abnormal minimizers. SIAM J Control Optim 32(6):1605–1620MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Moser J (1980) Geometry of quadrics and spectral theory. In: The Chern Symposium 1979. Proceedings of the International symposium on Differential Geometry held in honor of S.S. Chern, Berkeley, California. Springer, pp 147–188Google Scholar
  23. 23.
    Perelomov AM (1990) Integrable systems of classical mechanics and Lie algebras. Birkhauser, BaselCrossRefGoogle Scholar
  24. 24.
    Pontryagin LS, Boltyanski VG, Gamkrelidze RV, Mischenko EF (1962) The mathematical theory of optimal processes. Wiley, New YorkMATHGoogle Scholar
  25. 25.
    Reiman AG, Semenov Tian-Shansky MA (1994) Group-theoretic methods in the theory of finite dimensional integrable systems. In: Arnold VI, Novikov SP (eds) Encyclopedia of Mathematical Sciences. Springer, HeidelbergGoogle Scholar
  26. 26.
    Sussmann HJ, Willems J (2002) The brachistochrone problem and modern control theory. In: Anzaldo-Meneses A, Bonnard B, Gauthier J-P, Monroy-Perez F (eds) Contemporary trends in non-linear geometric control theory and its applications. Proceedings of the conference on Geometric Control Theory and Applications held in Mexico City on September 4–6, 2000, to celebrate the 60th anniversary of Velimir Jurdjevic. World Scientific Publishers, Singapore, pp 113–165Google Scholar
  27. 27.
    Sussmann HJ () Set separation, approximating multicones and the Lipschitz maximum principle. J Diff Equations (to appear)Google Scholar
  28. 28.
    Vinter RB (2000) Optimal control. Birkhauser, BostonMATHGoogle Scholar
  29. 29.
    Young LC (1969) Lectures in the calculus of variations and optimal control. Saunders, PhiladelphiaGoogle Scholar
  30. 30.
    Zelikin MI, Borisov VF (1994) Theory of chattering control with applications to aeronautics, Robotics, Economics, and Engineering. Birkhauser, BaselCrossRefGoogle Scholar

Books and Reviews

  1. 31.
    Bressan A (1985) A high-order test for optimality of bang-bang controls. SIAM J Control Optim 23:38–48MathSciNetMATHCrossRefGoogle Scholar
  2. 32.
    Griffiths P (1983) Exterior differential systems and the calculus of variations. Birkhauser, BostonMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Velimir Jurdjevic
    • 1
  1. 1.University of TorontoTorontoCanada