Skip to main content
SpringerLink
Log in

Controllability for Semilinear Functional and Neutral Functional Evolution Equations with Infinite Delay in Fréchet Spaces

  • Published:
Applied Mathematics and Optimization Submit manuscript

Abstract

The controllability of mild solutions defined on the semi-infinite positive real interval for two classes of first order semilinear functional and neutral functional differential evolution equations with infinite delay is studied in this paper. Our results are obtained using a recent nonlinear alternative due to Avramescu for sum of compact and contraction operators in Fréchet spaces, combined with the semigroup theory.

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

Access this article

Log in via an institution

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. Ahmed, N.U.: Semigroup Theory with Applications to Systems and Control. Pitman Research Notes in Mathematics Series, vol. 246. Longman Scientific & Technical/Wiley, New York (1991)

    MATH  Google Scholar 

  2. Ahmed, N.U.: Dynamic Systems and Control with Applications. World Scientific, Singapore (2006)

    MATH  Google Scholar 

  3. Arara, A., Benchohra, M., Górniewicz, L., Ouahab, A.: Controllability results for semilinear functional differential inclusions with unbounded delay. Math. Bull. 3, 157–183 (2006)

    MATH  Google Scholar 

  4. Avramescu, C.: Some remarks on a fixed point theorem of Krasnoselskii. Electron. J. Qual. Theory Differ. Equ. 5, 15 (2003)

    Google Scholar 

  5. Baghli, S., Benchohra, M.: Uniqueness results for partial functional differential equations in Fréchet spaces. Fixed Point Theory 9(2), 395–406 (2008)

    MATH  MathSciNet  Google Scholar 

  6. Baghli, S., Benchohra, M.: Existence results for semilinear neutral functional differential equations involving evolution operators in Fréchet spaces. Georgian Math. J. (to appear)

  7. Balachandran, K., Dauer, J.P.: Controllability of nonlinear systems in Banach spaces: A survey. Dedicated to Professor Wolfram Stadler. J. Optim. Theory Appl. 115, 7–28 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bensoussan, A., Da Prato, G., Delfour, M.C., Mitter, S.K.: Representation and Control of Infinite Dimension Systems. Systems and Control: Foundations and Applications, vol. 2. Birkhauser, Boston (1993)

    Google Scholar 

  9. Benchohra, M., Gatsori, E.P., Górniewicz, L., Ntouyas, S.K.: Controllability results for evolution inclusions with non-local conditions. Z. Anal. Anwend. 22(2), 411–431 (2003)

    MATH  Google Scholar 

  10. Benchohra, M., Górniewicz, L., Ntouyas, S.K., Ouahab, A.: Controllability results for nondensely semilinear functional differential equations. Z. Anal. Anwend. 25, 311–325 (2006)

    MATH  MathSciNet  Google Scholar 

  11. Benchohra, M., Górniewicz, L., Ntouyas, S.K.: Controllability of Some Nonlinear Systems in Banach spaces: The Fixed Point Theory Approach. Pawel Wlodkowicz University College, Plock (2003)

    MATH  Google Scholar 

  12. Benchohra, M., Górniewicz, L., Ntouyas, S.K.: Controllability of neutral functional differential and integrodifferential inclusions in Banach spaces with nonlocal conditions. Nonlinear Anal. Forum 7, 39–54 (2002)

    MATH  MathSciNet  Google Scholar 

  13. Benchohra, M., Górniewicz, L., Ntouyas, S.K.: Controllability results for multivalued semilinear differential equations with nonlocal conditions. Dyn. Syst. Appl. 11, 403–414 (2002)

    MATH  Google Scholar 

  14. Benchohra, M., Ntouyas, S.K.: Controllability results for multivalued semilinear neutral functional equations. Math. Sci. Res. J. 6, 65–77 (2002)

    MATH  MathSciNet  Google Scholar 

  15. Burton, T.A.: A fixed-point theorem of Krasnoselskii. Appl. Math. Lett. 11(1), 85–88 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Burton, T.A., Kirk, C.: A fixed point theorem of Krasnoselskii type. Math. Nachr. 189, 23–31 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chukwu, E.N., Lenhart, S.M.: Controllability questions for nonlinear systems in abstract spaces. J. Optim. Theory Appl. 68(3), 437–462 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  18. Curtain, R., Zwart, H.J.: An Introduction to Infinite Dimensional Linear Systems Theory. Springer, New York (1995)

    MATH  Google Scholar 

  19. Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)

    MATH  Google Scholar 

  20. Freidman, A.: Partial Differential Equations. Holt, Rinehat and Winston, New York (1969)

    Google Scholar 

  21. Fu, X.: Controllability of neutral functional differential systems in abstract space. Appl. Math. Comput. 141, 281–296 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Fu, X.: Controllability of abstract neutral functional differential systems with unbounded delay. Appl. Math. Comput. 151, 299–314 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gastori, E.P.: Controllability results for nondensely defined evolution differential inclusions with nonlocal conditions. J. Math. Anal. Appl. 297, 194–211 (2004)

    Article  MathSciNet  Google Scholar 

  24. Hale, J., Kato, J.: Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21, 11–41 (1978)

    MATH  MathSciNet  Google Scholar 

  25. Hino, Y., Murakami, S., Naito, T.: Functional Differential Equations with Unbounded Delay. Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991)

    Google Scholar 

  26. Kwun, Y.C., Park, J.Y., Ryu, J.W.: Approximate controllability and controllability for delay Volterra system. Bull. Korean Math. Soc. 28(2), 131–145 (1991)

    MATH  MathSciNet  Google Scholar 

  27. Lasiecka, L., Triggiani, R.: Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems. Appl. Math. Optim. 23, 109–154 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  28. Li, G., Song, Sh., Wu, Ch.: Controllability of evolution inclusions with nonlocal conditions. J. Syst. Sci. Complex. 18(1), 35–42 (2005)

    MATH  MathSciNet  Google Scholar 

  29. Li, G., Xue, X.: Controllability of evolution inclusions with nonlocal conditions. Appl. Math. Comput. 141, 375–384 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  30. Li, X., Yong, J.: Optimal Control Theory for Infinite Dimensional Systems. Birkhauser, Basel (1995)

    Google Scholar 

  31. Manitius, A., Triggiani, R.: Function space controllability of linear retarded systems: a derivation from abstract operator conditions. SIAM J. Control Optim. 16, 599–645 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  32. Manitius, A., Triggiani, R.: Sufficient conditions for function space controllability and feedback stabilizability for linear retarded systems. IEEE Trans. Automat. Control, 659–665 (1978)

  33. Naito, K.: On controllability for a nonlinear Volterra equation. Nonlinear Anal. 18(1), 99–108 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  34. Nakagri, S., Yamamoto, R.: Controllability and observability for linear retarded systems in Banach space. Int. J. Control 49, 1489–1504 (1989)

    Google Scholar 

  35. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    MATH  Google Scholar 

  36. Quinn, M.D., Carmichael, N.: An approach to nonlinear control problems using the fixed point methods, degree theory and pseudo-inverses. Numer. Funct. Anal. Optim. 7, 197–219 (1984–1985)

    Article  MathSciNet  Google Scholar 

  37. Triggiani, R.: On the stabilizability problem in Banach space. J. Math. Anal. Appl. 52(3), 383–403 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  38. Yosida, K.: Functional Analysis, 6th edn. Springer, Berlin (1980)

    MATH  Google Scholar 

  39. Zabczyk, J.: Mathematical Control Theory. Birkhauser, Basel (1992)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Florida Institute of Technology, Melboune, FL, 32901-6975, USA

    Ravi P. Agarwal

  2. Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, BP 89, 22000, Sidi Bel-Abbès, Algérie

    Selma Baghli & Mouffak Benchohra

Authors
  1. Ravi P. Agarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Selma Baghli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mouffak Benchohra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ravi P. Agarwal.

Additional information

Communicated by Roberto Triggiani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Agarwal, R.P., Baghli, S. & Benchohra, M. Controllability for Semilinear Functional and Neutral Functional Evolution Equations with Infinite Delay in Fréchet Spaces. Appl Math Optim 60, 253–274 (2009). https://doi.org/10.1007/s00245-009-9073-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-009-9073-1

Keywords

Search

Navigation