Advertisement

On the Existence of Solutions of Two Optimization Problems

  • Mariam Arabyan
Article
  • 168 Downloads

Abstract

In this paper, we prove the existence of solutions for the minimization problem of the shell weight for a given minimal frequency of the shell vibrations as well as for the maximization problem of the minimal frequency for a given shell weight. We consider an optimal control problem governed by an eigenvalue problem for a system of differential equations with variable coefficients. The form of the shell is considered as a control. Some of the coefficients are non-measurable. Earlier, we introduced certain special weighted functional spaces. By using these spaces, we establish the continuity of the considered minimal frequency functional and obtain the existence of solutions of both optimal control problems. At the end, we prove the Lipschitz continuity of the eigenvalue problem.

Keywords

Eigenvalues Eigenfunctions Continuous dependence The shell of rotation Weighted spaces Non-measurable coefficients Lipschitz continuity 

Mathematics Subject Classification

34B09 49J15 

Notes

Acknowledgements

We would like to thank the referees for their very thorough work and their many constructive comments.

References

  1. 1.
    Keller, J.B.: The shape of the strongest column. Arch. Ration. Mech. Anal. 5, 275–285 (1960)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Galilei, G.: Discorsi e dimostrazioni matematiche. Appresso gli Elsevirii, Leiden (1638)Google Scholar
  3. 3.
    Arabyan, M.H.: On an eigenvalue problem. YSU Sci. Notes 3, 31–39 (2005). (in Russian) zbMATHGoogle Scholar
  4. 4.
    Bramble, J.H., Osborn, J.E.: Rate of convergence estimates for nonselfadjoint approximations. Math. Comput. 27, 525–549 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Birkhauser, Boston (2005)zbMATHGoogle Scholar
  6. 6.
    He, Y., Guo, B.Z.: The existence of optimal solution for a shape optimization problem on starlike domain. J. Optim. Theory Appl. 152, 21–30 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Neittaanmaki, P., Sprekels, J., Tiba, D.: Optimization of Elliptic Systems. Springer, New York (2006)zbMATHGoogle Scholar
  8. 8.
    Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  9. 9.
    Velichkov, B.: Existence and Regularity Results for Some Shape Optimization Problems. Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  10. 10.
    Wang, G., Wang, L., Yang, D.: Shape optimization of an elliptic equation in an exterior domain. SIAM J. Control Optim. 45, 532–547 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zaslavski, A.J.: Nonconvex Optimal Control and Variational Problems. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  12. 12.
    Arabyan, M.H.: The smoothness of generalized waveforms for the problem of rotation of the shell oscillations depending on certain nonsummable coefficients. Izv. NAS Armen. Mech. 69, 28–40 (2016). (in Russian) MathSciNetGoogle Scholar
  13. 13.
    Zaslavski, A.J.: Structure of Approximate Solutions of Optimal Control Problems. Springer International Publishing, New York (2013)CrossRefzbMATHGoogle Scholar
  14. 14.
    Carlson, Dean A.: The existence of optimal control for problems defined on time scales. J. Optim. Theory Appl. 166, 351–376 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Timoshenko, S.: Strength of Materials. Part 1: Elementary Theory and Problems. Krieger Publishing Company, Melbourne (1976)Google Scholar
  16. 16.
    Timoshenko, S.: Strength of Materials. Part 2: Advanced Theory and Problems. Krieger Publishing Company, Melbourne (1976)zbMATHGoogle Scholar
  17. 17.
    Flugge, W.: Statik und Dynamik der Schalen. Springer, Berlin (1962)CrossRefzbMATHGoogle Scholar
  18. 18.
    Grigolyuk, E.I.: Nonlinear oscillations and stability of shallow rods and shells. Izv. AS USSR Dep. Tech. Sci. 3, 33–98 (1955). (in Russian) Google Scholar
  19. 19.
    Kolmogorov, A.N., Fomin, S.V.: The Elements of the Theory of Functions and Functional Analysis. Dover Publications, New York (1999)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Yerevan State UniversityYerevanArmenia

Personalised recommendations