Skip to main content
SpringerLink
Log in

The Optimal Vector Control for the Elastic Oscillations Described by Fredholm Integral-Differential Equations

  • Conference paper
  • First Online:
Analysis and Partial Differential Equations: Perspectives from Developing Countries

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 275))

Abstract

In this paper, we investigate the nonlinear problem of the optimal vector control for oscillation processes described by Fredholm integro-differential equations in partial derivatives when function of external sources nonlinearly depend on control parameters. It was found that the system of nonlinear integral equations,which obtained relatively to the components of the optimal vector control, have the property of equal relations. This fact lets us to simplify the procedure of the constructing the solution of the nonlinear optimization problem. We have developed algorithm for constructing the solution of the nonlinear optimization problem.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Volterra, V.: Theory of Functionals and of Integral and Integro-Differential Equations. New York, USA (2005)

    Google Scholar 

  2. Vladimirov, V.S.: Mathematical problems of speed transport theory of particles, pp. 3–158. Trudy MIAN, T. 61 (1961) (In Russian)

    Google Scholar 

  3. Egorov, A.I.: Optimal control of thermal and diffusion processes. M: Nauka (1978) (In Russian)

    Google Scholar 

  4. Egorov, A.I.: On optimal control for processes in distributed objects. Appl. Math. Mech. VII(4), 688–696 (1963) (In Russian)

    Article  MathSciNet  Google Scholar 

  5. Egorov, A.I.: On optimal control for processes in some systems with distributed parameters. In: Avtomatika and Telemechanika, vol. XXV, No. 5 (1964) (In Russian)

    Google Scholar 

  6. Sirazetdinov, T.K.: To the theory of optimal processes with distributed parameters. In: Avtomatika and Telemechanika, vol. XXV, No. 4 (1964) (In Russian)

    Google Scholar 

  7. Egorov, A.I.: Optimal processes and invariance theory. SIAM J. Control 4(4), 601–661 (1966)

    Article  MathSciNet  Google Scholar 

  8. Butkovskii, A.G., Egorov, A.I., Lurie, A.K.: Optimal control of distributed systems. SIAM J. Control 6(13), 437–476 (1968)

    Article  MathSciNet  Google Scholar 

  9. Wanfang, Sh., Danping, Y., Wenbin, L.: Optimal control problem governed by a linear hyperbolic integro-differential equation and its finite element analysis. Bound. Value Probl. 2014(173) (2014)

    Google Scholar 

  10. Kowalewski, A.: Optimal control of an infinite order hyperbolic system with multiple time-varying lags. Automatyka, Tom 15, 53–65 (2011)

    Google Scholar 

  11. Asatur, zh. K.:On optimal boundary and distributed control of partial integro-differential equations. Arch. Control. Sci. 24(LX)(1), 5–25 (2014)

    Google Scholar 

  12. Appel, J.M., Kalitvin, A.S., Zabrejko, P.P.: Partial Integral Operators and Integro-Differential Equations, IM. Dekkar, New York (2000)

    Google Scholar 

  13. Sachs, E.W., Strauss, A.K.: Efficient solution of partial integrodifferential equation in finance. Appl. Numer. Math. 58(11), 1687–1703 (2008)

    Article  MathSciNet  Google Scholar 

  14. Thorwe, J., Bhalekar, S.: Solving partial integrodifferential equations using Laplace transform method. Am. J. Comput. Appl. Math. 2(3), 101–104 (2012)

    Article  Google Scholar 

  15. Kerimbekov, A.: On the solvability of a nonlinear optimal control problem for the thermal processes described by fredholm integro-differential equations. In: Proceedings of the 9th ISAAC Congress Current Trends in Analysis and its Applications (Krakov 2013). A series of trends in mathematics, vol. XVI, pp. 803–811. Springer International Publishing, Switzerland (2015)

    Google Scholar 

  16. Komkov, V.: Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems, pp. 160. Springer (1972)

    Google Scholar 

  17. Kerimbekov, A., Baetov, A.K.: On solvability of a nonlinear optimization problem for oscillation processes. In: Proceedings of International Scientific conference Actual Problems of Applied Mathematics and Information Technologies-al-Horezmi. Tashkent (Uzbekistan), pp. 301–303 (2012) (In Russian)

    Google Scholar 

  18. Kerimbekov, A., Kabaeva, Z.C.: Solution of the optimization problem for thermal processes in nonlinear vector control. In: Proceedings of the VI International Scientific Conference. Part I, Aktobe Regional University K.Zhubanova. Aktobe (Kazakhstan), 14-17 Oct 2012, pp. 100–104 (2012) (In Russian)

    Google Scholar 

  19. Tikhonov, A.N., Samarskiy, A.A.: Equations of Mathematical Physics. M. Nauka, Moscow (1972) (In Russian)

    Google Scholar 

  20. Plotnikov, V.I.: The energy inequality and over-determination property of systems of eigen function. In: The News of Academy of Science of USSR, vol. 32, No. 4, pp. 743–755 (Math. Ser.) (1968)

    Google Scholar 

  21. Tricomi, I.F.: Integral Equations. Intersciense Publishers, New York (1957)

    MATH  Google Scholar 

  22. Krasnov, M.V.: Integral Equations, pp. 303. Nauka, Moscow (1975) (In Russian)

    Google Scholar 

  23. Lusternik, L.A., Sobolev, V.I.: Elements of Functional Analysis, pp. 520. Nauka, Moscow (1965) (In Russian)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kyrgyz-Turkish “Manas” University, Bishkek, Kyrgyzstan

    Elmira Abdyldaeva

  2. Department of Applied Mathematics and Informatics, Kyrgyz-Russian Slavic University, Bishkek, Kyrgyzstan

    Akylbek Kerimbekov

Authors
  1. Elmira Abdyldaeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akylbek Kerimbekov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elmira Abdyldaeva .

Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Queen Mary University of London, London, UK

    Julio Delgado

  2. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium

    Michael Ruzhansky

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Abdyldaeva, E., Kerimbekov, A. (2019). The Optimal Vector Control for the Elastic Oscillations Described by Fredholm Integral-Differential Equations. In: Delgado, J., Ruzhansky, M. (eds) Analysis and Partial Differential Equations: Perspectives from Developing Countries. Springer Proceedings in Mathematics & Statistics, vol 275. Springer, Cham. https://doi.org/10.1007/978-3-030-05657-5_3

Download citation

Publish with us

Policies and ethics

Search

Navigation