Advertisement

Afrika Matematika

, Volume 29, Issue 1–2, pp 33–46 | Cite as

A formulation of Noether’s theorem for fuzzy problems of the calculus of variations

  • O. S. Fard
  • J. Soolaki
  • R. Almeida
Article
  • 46 Downloads

Abstract

The theory of the calculus of variations for fuzzy systems was recently initiated in Farhadinia (Inf Sci 181:1348–1357, 2011), with the proof of the fuzzy Euler–Lagrange equation. Using fuzzy Euler–Lagrange equation, we obtain here a Noether–like theorem for fuzzy variational problems.

Keywords

Fuzzy Noether’s theorem Fuzzy Euler–Lagrange conditions Fuzzy conservation law 

Mathematics Subject Classification

Primary 93C42 Secondary 34N05 93D05 

Notes

Acknowledgements

R. Almeida was supported by Portuguese funds through the CIDMA-Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013.

References

  1. 1.
    Brunt, B.V.: The Calculus of Variations. Springer, Heidelberg (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Buckley, J.J., Feuring, T.: Introduction to fuzzy partial differential equations. Fuzzy Sets Syst. 105, 241–248 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kosmann-Schwarzbach, Y.: Les théorèmes de Noether, 2nd edn. With a translation of the original article Invariante Variations probleme, Editions de l Ecole Polytechnique, Palaiseau (2006)Google Scholar
  4. 4.
    Torres, D.F.M.: Quasi-invariant optimal control problems. Port. Math. (N.S.) 61(1), 97–114 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Torres, D.F.M.: Proper extensions of Noethers symmetry theorem for nonsmooth extremals of the calculus of variations. Commun. Pure Appl. Anal. 3(3), 491–500 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Frederico, G.S.F., Torres, D.F.M.: A formulation of Noethers theorem for fractional problems of the calculus of variations. J. Math. Anal. Appl. 334, 834–846 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hoa, N.V.: Fuzzy fractional functional differential equations under Caputo gH-differentiability. Commun. Nonlinear Sci. Numer. Simul. 22, 1134–1157 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Farhadinia, B.: Necessary optimality conditions for fuzzy variational problems. Inf. Sci. 181, 1348–1357 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fard, O.S., Zadeh, M.S.: Note on “Necessary optimality conditions for fuzzy variational problems”. J. Adv. Res. Dyn. Control Syst. 4(3), 1–9 (2012)MathSciNetGoogle Scholar
  10. 10.
    Fard, O.S., Borzabadi, A.H., Heidari, M.: On fuzzy Euler-Lagrange equations. Ann. Fuzzy Math. Inform. 7(3), 447–461 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Goetschel Jr., R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets Syst. 18(1), 31–43 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Xu, J., Liao, Z., Nieto, J.J.: A class of linear differential dynamical systems with fuzzy matrices. J. Math. Anal. Appl. 368(1), 54–68 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceDamghan UniversityDamghanIran
  2. 2.Department of MathematicsDamghan UniversityDamghanIran
  3. 3.Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA)University of AveiroAveiroPortugal

Personalised recommendations