[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The present paper extends the classical second-order variational problem of Herglotz type to the more general context of the Euclidean sphere \(S^n\) following variational and optimal control approaches. The relation between the Hamiltonian equations and the generalized Euler–Lagrange equations is established. This problem covers some classical variational problems posed on the Riemannian manifold \(S^n\) such as the problem of finding cubic polynomials on \(S^n\). It also finds applicability on the dynamics of the simple pendulum in a resistive medium.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Vujanovic, B.D., Jones, S.E.: Variational Methods in Nonconservative Phenomena. Mathematics in Science and Engineering, vol. 182. Academic Press Inc., Cambridge (1989)

    MATH  Google Scholar 

  2. Herglotz, G.: Berührungstransformationen. Lectures at the University of Göttingen, Göttingen (1930)

  3. Georgieva, B., Guenther, R.: First Noether-type theorem for the generalized variational principle of Herglotz. Topol. Methods Nonlinear Anal. 20(2), 261–273 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Guenther, R., Gottsch, J.A., Kramer, D.B.: The Herglotz algorithm for constructing canonical transformations. SIAM Rev. 38(2), 287–293 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Guenther, R., Guenther, C.M., Gottsch, J.A.: The Herglotz Lectures on Contact Transformations and Hamiltonian Systems. Lecture Notes in Nonlinear Analysis, vol. 1. Juliusz Schauder Center for Nonlinear Studies, Nicholas Copernicus University, Torún (1996)

    MATH  Google Scholar 

  6. Georgieva, B.: Symmetries of the Herglotz variational principle in the case of one independent variable. Ann. Sofia Univ. Fac. Math. Inf. 100, 113–122 (2010)

    MathSciNet  Google Scholar 

  7. Georgieva, B., Guenther, R.: Second Noether-type theorem for the generalized variational principle of Herglotz. Topol. Methods Nonlinear Anal. 26(2), 307–314 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Georgieva, B., Guenther, R., Bodurov, T.: Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem. J. Math. Phys. 44(9), 3911–3927 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Santos, S.P.S., Martins, N., Torres, D.F.M.: Variational problems of Herglotz type with time delay: Dubois–Reymond condition and Noether’s first theorem. Discret. Contin. Dyn. Syst. Ser. A 35(9), 4593–4610 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Santos, S.P.S., Martins, N., Torres, D.F.M.: An optimal control approach to Herglotz variational problems. Optimization in the natural sciences. Commun. Comput. Inf. Sci. (CCIS) 449, 107–117 (2015)

    Google Scholar 

  11. Santos, S.P.S., Martins, N., Torres, D.F.M.: Noether’s theorem for higher-order variational problems of Herglotz type. In: 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, vol. 2015, pp. 990–999 (2015)

  12. Santos, S.P.S., Martins, N., Torres, D.F.M.: Higher-order variational problems of Herglotz type with time delay. Pure Appl. Funct. Anal. 1(2), 291–307 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Santos, S.P.S., Martins, N., Torres, D.F.M.: Noether currents for higher-order variational problems of Herglotz type with time delay. Discret. Contin. Dyn. Syst. Ser. S 11(1), 91–102 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Zhang, Y.: Variational problem of Herglotz type for Birkhoffian system and its Noether’s theorems. Acta Mech. 228(4), 1481–1492 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tian, X., Zhang, Y.: Noether’s theorem and its inverse of Birkhoffian system in event space based on Herglotz variational problem. Int. J. Theor. Phys. 57(3), 887–897 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Birkhoff, G.D.: Dynamical Systems. AMS College Publication, Providence (1927)

    MATH  Google Scholar 

  17. Santilli, R.M.: Foundations of Theoretical Mechanics II. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  18. Almeida, R., Malinowska, A.B.: Fractional variational principle of Herglotz. Discret. Contin. Dyn. Syst. Ser. B 19(8), 2367–2381 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)

    Article  MathSciNet  Google Scholar 

  20. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3582–3592 (1997)

    Article  MathSciNet  Google Scholar 

  21. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Agrawal, O.P., Baleanu, D.: Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13(9–10), 1269–1281 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jarad, F., Abdeljawad, T., Baleanu, D.: Fractional variational optimal control problems with delayed arguments. Nonlinear Dyn. 62, 609–614 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jarad, F., Abdeljawad, T., Baleanu, D.: Higher order fractional variational optimal control problems with delayed arguments. Appl. Math. Comput. 218, 9234–9240 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Bahaa, G.M.: Fractional optimal control problem for differential system with delay argument. Adv. Differ. Equ. 69, 1–19 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Abrunheiro, L., Machado, L., Martins, N.: The Herglotz variational problem on spheres and its optimal control approach. J. Math. Anal. 7(1), 12–22 (2016)

    MathSciNet  MATH  Google Scholar 

  27. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60, 2nd edn. Springer, Berlin (1989)

    Book  Google Scholar 

  28. Bloch, A.M, Crouch, P.E.: Reduction of Euler–Lagrange problems for constrained variational problems and relation with optimal control problems. In: Proceedings of the 33rd Conference on Decision and Control, Orlando, FL, pp. 2584–2590 (1994)

  29. Bloch, A.M., Crouch, P.E.: Nonholonomic control systems on Riemannian manifolds. SIAM J. Control Optim. 33, 126–148 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Bloch, A.M., Crouch, P.E.: On the equivalence of higher order variational problems and optimal control problems. In: Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, pp. 1648–1653 (1996)

  31. do Carmo, M.P.: Riemannian Geometry. Mathematics: Theory and Applications. Birkäuser, Boston (1992)

    Book  Google Scholar 

  32. Jost, J.: Riemannian Geometry and Geometric Analysis. Universitext, 6th edn. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  33. de León, M., Rodrigues, P.R.: Generalized Classical Mechanics and Field Theory: A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives. North-Holland Mathematics Studies, Nachbin, L. (ed.), vol. 112. Elsiever, Amsterdam (1985)

    Google Scholar 

  34. Santos, S.P.S., Martins, N., Torres, D.F.M.: Higher-order variational problems of Herglotz type. Vietnam J. Math. 42, 409–419 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Bonnard, B., Faubourg, L., Launay, G., Trélat, E.: Optimal control with state constraints and the space shuttle re-entry problem. J. Dyn. Control Syst. 9(2), 155–199 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Jacobson, D.H., Lee, M.M.: New necessary conditions of optimality for control problems with state-variable inequality constraints. J. Math. Anal. Appl. 35, 255–284 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  37. Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved spaces. IMA J. Math. Control Inf. 6, 465–473 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. Abrunheiro, L. Camarinha, M., Clemente-Gallardo, J.: Geometric Hamiltonian formulation of a variational problem depending on the covariant acceleration. In: The Cape Verde International Days on Mathematics 2013, Praia, Cape Verde, Conference Papers in Mathematics, pp. 1–9. Hindawi Publishing Corporation (2013)

  39. Mickens, R.E.: An Introduction to Nonlinear Oscillations. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  40. Nelson, R.A., Olsson, M.G.: The pendulum: Rich physics from a simple system. Am. J. Phys. 54(2), 112–121 (1986)

    Article  Google Scholar 

  41. Camarinha, M., Silva Leite, F., Crouch, P.: Elastic curves as solutions of Riemannian and sub-Riemannian control problems. Math. Control Signal Syst. 13(2), 140–155 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work of Lígia Abrunheiro and Natália Martins was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project UID/MAT/04106/2013. Luís Machado acknowledges “Fundação para a Ciência e a Tecnologia” (FCT–Portugal) and COMPETE 2020 Program for financial support through project UID-EEA-00048-2013.

The authors would like to thank the reviewers for their valuable suggestions to improve the quality of the paper.

Author information

Authors and Affiliations

  1. Institute of Systems and Robotics, University of Coimbra, Coimbra, Portugal

    Luís Machado

  2. Department of Mathematics, University of Trás-os-Montes e Alto Douro (UTAD), Vila Real, Portugal

    Luís Machado

  3. Center for Research and Development in Mathematics and Applications (CIDMA), Higher Institute of Accounting and Administration, University of Aveiro, Aveiro, Portugal

    Lígia Abrunheiro

  4. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Natália Martins

Authors
  1. Luís Machado
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lígia Abrunheiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Natália Martins
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luís Machado.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Machado, L., Abrunheiro, L. & Martins, N. Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres. J Optim Theory Appl 182, 965–983 (2019). https://doi.org/10.1007/s10957-018-1424-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-018-1424-0

Keywords

Mathematics Subject Classification

Search

Navigation