Skip to main content
Log in

Nonsmooth integral directing functions in the problems of forced oscillations

  • Nonlinear Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

The method of directing function was extended to the nonsmooth case. It is used to solve the problem of periodic oscillations of the control plants obeying the functional-differential inclusions whose right side is not convex-valued and satisfies the condition for almost semicontinuity from below.

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

Access this article

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. Myshkis, A.D., General Theory of Differential Delay Equations, Usp. Mat. Nauk, 1949, vol. 4, no. 5, pp. 99–141.

    MATH  Google Scholar 

  2. Krasnoselskii, A.M., Krasnoselskii, M.A., Mawhin, J., and Pokrovskii, A., Generalized Guiding Functions in a Problem on High Frequency Forced Oscillations, Nonlin. Anal. Theory, Methods Appl., 1994, vol. 22, no. 11, pp. 1357–1371.

    Article  MathSciNet  Google Scholar 

  3. Krasnosel’skii, M.A., Operator sdviga po traektoriyam differentsial’nykh uravnenii (Operator of Shift Along the Trajectories of Differential Equations) Moscow: Nauka, 1966.

    Google Scholar 

  4. Krasnosel’skii, M.A. and Perov, A.I., On One Principle of Existence of Bounded Periodic and Almostperiodic Solutions of Systems of Ordinary Differential Equations, Dokl. Akad. Nauk SSSR, 1958, vol. 123, no. 2, pp. 235–238.

    MathSciNet  Google Scholar 

  5. Borisovich, Yu.G., Gel’man, B.D., Myshkis, A.D., and Obukhovskii, V.V., Vvedenie v teoriyu mnogoznachnykh otobrazhenii i differentsial’nykh vklyuchenii (Introduction to the Theory of Multivalued Maps and Differential Inclusions), Moscow: “Librokom,” 2011.

    Google Scholar 

  6. Górniewicz, L., Topological Fixed Point Theory of Multivalued Mappings, Dordrecht: Kluwer, 1999.

    Book  MATH  Google Scholar 

  7. Fonda, A., Guiding Functions and Periodic Solutions to Functional Differential Equations, Proc. Am. Math. Soc., 1987, vol. 99, no. 1, pp. 79–85.

    MATH  MathSciNet  Google Scholar 

  8. Rachinskii, D.I., Forced Oscillations in Control Systems under Near-Resonance Conditions, Autom. Remote Control, 1995, vol. 56, no. 11, part 1, pp. 1575–1584.

    MathSciNet  Google Scholar 

  9. Kornev, S.V. and Obukhovskii, V.V., On Integral Directing Functions for Functional-Differential Inclusions, in Topologicheskie metody nelineinogo analiza (Topological Methods of Nonlinear Analysis), Voronezh: Voronezh. Gos. Univ., 2000, pp. 87–107.

    Google Scholar 

  10. Kornev, S.V., On the Method of Multivalent Guiding Functions for Periodic Solutions of Differential Inclusions, Autom. Remote Control, 2003, vol. 64, no. 3, pp. 409–419.

    Article  MATH  MathSciNet  Google Scholar 

  11. De Blasi, F.S., Górniewicz, L., and Pianigiani, G., Topological Degree and Periodic Solutions of Differential Inclusions, Nonlin. Anal., 1999, vol. 37, pp. 217–245.

    Article  MATH  Google Scholar 

  12. Kornev, S.V. and Obukhovskii, V.V., On Nonsmooth Multivalent Directing Functions, Differ. Uravn., 2003, vol. 39, no. 11, pp. 1497–1502.

    MathSciNet  Google Scholar 

  13. Kornev, S. and Obukhovskii, V., On Some Developments of the Method of Integral Guiding Functions, Differ. Uravn., 2003, vol. 12, nos. 3–4, pp. 303–310.

    MathSciNet  Google Scholar 

  14. Obukhovskii, V., Zecca, P., Loi, N.V., and Kornev, S., Method of Guiding Functions in Problems of Nonlinear Analysis, Lecture Notes Math., vol. 2076, Berlin: Springer, 2013.

  15. Kamenskii, M., Obukhovskii, V., and Zecca, P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Berlin: Walter de Gruyter, 2001.

    Book  MATH  Google Scholar 

  16. Bressan, A. and Colombo, G., Extensions and Selections of Maps with Decomposable Values, Studia Math., 1988, vol. 90, pp. 69–86.

    MATH  MathSciNet  Google Scholar 

  17. Fryszkowski, A., Fixed Point Theory for Decomposable Sets, Dordrecht: Kluwer, 2004.

    Book  MATH  Google Scholar 

  18. Emel’yanov, S.V., Korovin, S.K., Bobylev, N.A., and Bulatov, A.V., Gomotopii ekstremal’nykh zadach (Homotopies of Extremal Problems), Moscow: Nauka, 2001.

    Google Scholar 

  19. Clarke, F.H., Optimization and Nonsmooth Analysis, New York: Wiley, 1983. Translated under the title Optimizatsiya i negladkii analiz, Moscow: Nauka, 1988.

    MATH  Google Scholar 

  20. Bader, R., Gel’man, B.D., and Obukhovskii, V.V., On a Class of Multivalued Maps, Vestn. Voronezh. Gos. Univ., Ser. Fiz.-Mat., 2003, vol. 2, pp. 35–38.

    Google Scholar 

  21. Michael, E., Continuous Selections. I, Ann. Math., 1956, vol. 63, pp. 361–382.

    Article  MATH  MathSciNet  Google Scholar 

  22. Kornev, S.V. and Obukhovskii, V.V., On Some Variants of the Topological Degree Theory for the Nonconvexvalued Multimaps and Periodic Problems for Functional-Differential Inclusion, Tr. Mat. Fak., Voronezh. Gos. Univ., 2004, vol. 8, pp. 56–75.

    Google Scholar 

  23. Kornev, S. and Obukhovskii, V., On Asymptotics of Solutions for a Class of Functional Differential Inclusions, Discuss. Math. Differ. Incl., Control Optim., 2014, vol. 34, no. 2, pp. 219–227.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. V. Kornev.

Additional information

Original Russian Text © S.V. Kornev, 2015, published in Avtomatika i Telemekhanika, 2015, No. 9, pp. 31–43.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kornev, S.V. Nonsmooth integral directing functions in the problems of forced oscillations. Autom Remote Control 76, 1541–1550 (2015). https://doi.org/10.1134/S0005117915090027

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117915090027

Keywords

Navigation