Skip to main content
SpringerLink
Log in

Presymplectic high order maximum principle

  • Preliminary Communication
  • Published:
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

Pontryagin’s Maximum Principle is an outstanding result for solving optimal control problems by means of optimizing a specific function on some particular variables, the so called controls. However, this is not always enough for solving all these problems. A high order maximum principle Krener (SIAM J Control Optim 15(2):256–293, 1977) must be used in order to obtain more necessary conditions for optimality. These new conditions determine candidates to be optimal controls for a wider range of optimal control problems. Here, we focus on control-affine systems. Krener’s high order perturbations are redefined following the notions introduced in Aguilar and Lewis (Proceedings of the 18th mathematical theory of networks and systems in Blacksburg, Virginia, 2008). A weaker version of Krener’s high order maximum principle is stated in the framework of presymplectic geometry. As a result, the presymplectic constraint algorithm in the sense of Gotay et al. (J Math Phys 19(11):2388–2399, 1978) can be used. We establish the connections between the presymplectic constraint algorithm and the candidates to be optimal curves obtained from the necessary conditions in Krener’s high order maximum principle. In this paper we obtain weaker geometric necessary conditions for optimality of abnormal solutions than the ones in Krener (SIAM J Control Optim 15(2):256–293, 1977) and the ones in the weak high order maximum principle. These new necessary conditions are more useful, computationally speaking, for finding curves candidate to be optimal. The theory is supported by describing specifically some of the above-mentioned conditions for some mechanical control systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agrachev, A., Sachkov, Y.L.: Control theory from the geometric viewpoint (Control Theory and Optimization, II). In: Encyclopaedia of Mathematical Sciences, vol. 87. Springer, Berlin (2004)

  2. Aguilar, C., Lewis, A.D.: A jet bundle setting for studying the reachable set. In: Proceedings of the 18th Mathematical Theory of Networks and Systems in Blacksburg (Virginia) (2008)

  3. Barbero-Liñán M., Muñoz Lecanda M.C.: Geometric approach to Pontryagin’s maximum principle. Acta Appl. Math. 108, 429–485 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barbero-Liñán M., Muñoz Lecanda M.C.: Constraint algorithm for extremals in optimal control problems. Int. J. Geom. Methods Mod. Phys. 6(7), 1221–1233 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bianchini R.M.: High order necessary optimality conditions, Control theory and its applications (Grado, 1998). Rend. Sem. Mat. Univ. Politec. Torino 56(4), 41–51 (2001)

    MathSciNet  Google Scholar 

  6. Bianchini R.M., Stefani G.: Controllability along a trajectory: a variational approach. SIAM J. Control Optim. 31(4), 900–927 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bullo, F., Lewis, A.D.: Geometric control of mechanical systems: modeling, analysis, and design for simple mechanical control systems. In: Texts in Applied Mathematics. vol. 49. Springer, New York (2005)

  8. Bullo, F., Lewis, A.D.: Supplementary Chapters of Geometric Control of Mechanical Systems. http://penelope.mast.queensu.ca/smcs/ (2005)

  9. Coddington E.A., Levinson N.: Theory of Ordinary Differential Esquations. McGraw-Hill, New York (1955)

    Google Scholar 

  10. Gabasov R., Kirillova F.M.: High order necessary conditions for optimality. SIAM J. Control 10, 127–168 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gotay M.J., Nester J.M., Hinds G.: Presymplectic manifolds and the Dirac–Bergmann theory of constraints. J. Math. Phys. 19(11), 2388–2399 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hermes H.: Local controllability and sufficient conditions in singular problems. J. Differ. Equ. 20(1), 213–232 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kawski, M.: High-order maximal principles. In: New trends in nonlinear dynamics and control, and their applications, Lecture Notes in Control and Inform. Sci., vol. 295, pp. 313–326. Springer, Berlin (2003)

  14. Knobloch, H.W.: Higher Order Necessary Conditions in Optimal Control Theory, Lecture Notes in Computer Science, vol. 34. Springer, Berlin (1981)

  15. Krener A.J.: The high order maximal principle and its application to singular extremals. SIAM J. Control Optim. 15(2), 256–293 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  16. López C., Martínez E.: Sub-Finslerian metric associated to an optimal control system. SIAM J. Control Optim. 39(3), 798–811 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, W., Sussmann, H.J.: Shortest paths for sub-Riemannian metrics on rank-two distributions. Memb. Am. Math. Soc., 118(564) (1995)

  18. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt. Interscience/Wiley, New York (1962)

  19. Saunders, D.J.: The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series, vol. 142. Cambridge University Press, Cambridge (1989)

  20. Strichartz R.: The Campbell–Baker–Hausdor–Dynkin formula and solutions of differential equations. J. Funct. Anal. 72, 320–345 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sussmann, H.J.: An introduction to the coordinate-free maximum principle. Geometry of Feedback and Optimal Control, Monogr. Textbooks Pure Appl. Math., vol. 207, pp. 463–557. Dekker, New York (1998)

Download references

Author information

Author notes

  1. M. Barbero-Liñán

    Present address: Instituto de Ciencias Mateḿaticas (CSIC-UAM-UC3M-UCM), Madrid, Spain

Authors and Affiliations

  1. Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, University Ave., Kingston, ON, K7L 3N6, Canada

    M. Barbero-Liñán

  2. Departamento de Matemática Aplicada IV, Edificio C-3, Campus Norte UPC, C/Jordi Girona 1, 08034, Barcelona, Spain

    M. C. Muñoz-Lecanda

Authors
  1. M. Barbero-Liñán
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. C. Muñoz-Lecanda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Barbero-Liñán.

Additional information

We acknowledge the financial support of Ministerio de Educación y Ciencia, Project MTM2008-00689/MTM and the Network Project MTM2008-03606-E/, and of the Catalan government, 2009SGR1338. The first author acknowledges the financial support of Comissionat per a Universitats i Recerca del Departament d’Innovació, Universitats i Empresa of Generalitat de Catalunya in the preparation of this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barbero-Liñán, M., Muñoz-Lecanda, M.C. Presymplectic high order maximum principle. RACSAM 106, 97–110 (2012). https://doi.org/10.1007/s13398-011-0022-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-011-0022-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation