iBet uBet web content aggregator. Adding the entire web to your favor.
iBet uBet web content aggregator. Adding the entire web to your favor.



Link to original content: https://unpaywall.org/10.1007/S10957-012-0174-7
Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators | Journal of Optimization Theory and Applications Skip to main content
Springer Nature Link
Log in

Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The paper is concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach spaces. With the help of two new characteristic solution operators and their properties, such as boundedness and compactness, we present the controllability results corresponding to two admissible control sets via the well-known Schauder fixed point theorem. Finally, an example is given to illustrate our theoretical results.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics. Springer, New York (2010)

    Book  MATH  Google Scholar 

  2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  3. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  4. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  5. Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media Springer, New York (2010)

    Google Scholar 

  6. El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433–440 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. El-Borai, M.M.: The fundamental solutions for fractional evolution equations of parabolic type. J. Appl. Math. Stoch. Anal. 3, 197–211 (2004)

    Article  MathSciNet  Google Scholar 

  8. Balachandran, K., Park, J.Y.: Controllability of fractional integrodifferential systems in Banach spaces. Nonlinear Anal. Hybrid Syst. 3, 363–367 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl. 11, 4465–4475 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hernández, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal., Theory Methods Appl. 73, 3462–3471 (2010)

    Article  MATH  Google Scholar 

  12. Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, 262–272 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang, J., Zhou, Y.: Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal., Real World Appl. 12, 3642–3653 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, J., Zhou, Y.: Analysis of nonlinear fractional control systems in Banach spaces. Nonlinear Anal., Theory Methods Appl. 74, 5929–5942 (2011)

    Article  MATH  Google Scholar 

  15. Sakthivel, R., Ren, Y., Mahmudov, N.I.: On the approximate controllability of semilinear fractional differential systems. Comput. Math. Appl. 62, 1451–1459 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Debbouchea, A., Baleanu, D.: Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62, 1442–1450 (2011)

    Article  MathSciNet  Google Scholar 

  17. Wang, J., Zhou, Y., Medved’, M.: On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay. J. Optim. Theory Appl. 152, 31–50 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, J., Zhou, Y.: Mittag-Leffer-Ulam stabilities of fractional evolution equations. Appl. Math. Lett. 25, 723–728 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, J., Zhou, Y., Wei, W.: Optimal feedback control for semilinear fractional evolution equations in Banach spaces. Syst. Control Lett. 61, 472–476 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, J., Fan, Z., Zhou, Y.: Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. J. Optim. Theory Appl. 154, 292–302 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, J., Zhou, Y.: Complete controllability of fractional evolution systems. Commun. Nonlinear Sci. Numer. Simul. 17, 4346–4355 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, J., Zhou, Y., Wei, W.: Fractional Schrödinger equations with potential and optimal controls. Nonlinear Anal., Real World Appl. 13, 2755–2766 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, R.N., Chen, D.H., Xiao, T.J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equs. 252, 202–235 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kumar, S., Sukavanam, N.: Approximate controllability of fractional order semilinear systems with bounded delay. J. Differ. Equs. 252, 6163–6174 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, K., Peng, J., Jia, J.: Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives. J. Funct. Anal. 263, 476–510 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Balachandran, K., Dauer, J.P.: Controllability of functional differential systems of Sobolev type in Banach spaces. Kybernetika 34, 349–357 (1998)

    MathSciNet  MATH  Google Scholar 

  27. Li, F., Liang, J., Xu, H.K.: Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions. J. Math. Anal. Appl. 391, 510–525 (2012)

    Article  MATH  Google Scholar 

  28. Lightbourne, J.H., Rankin, S.M.: A partial functional differential equation of Sobolev type. J. Math. Anal. Appl. 93, 328–337 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  29. Berberan-Santos, M.N.: Relation between the inverse Laplace transforms of I(t β) and I(t): application to the Mittag–Leffler and asymptotic inverse power law relaxation functions. J. Math. Chem. 38, 265–270 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Berberan-Santos, M.N.: Properties of the Mittag–Leffler relaxation function. J. Math. Chem. 38, 629–635 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the referees for their careful reading of the manuscript and insightful comments. We also acknowledge the valuable comments and suggestions from the editors. Finally, The first author acknowledges the support by Grants VEGA-MS 1/0507/11, VEGA-SAV 2/0124/12 and APVV-0414-07; the second author acknowledges the support by National Natural Science Foundation of China (11201091) and Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169) and the third author acknowledges the support by National Natural Science Foundation of China (11271309), Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001) and Key Projects of Hunan Provincial Natural Science Foundation of China (12JJ2001).

Author information

Authors and Affiliations

  1. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia

    Michal Fec̆kan

  2. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Michal Fec̆kan

  3. Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, P.R. China

    JinRong Wang

  4. Department of Mathematics, Xiangtan University, Xiangtan, Hunan, 411105, P.R. China

    Yong Zhou

Authors
  1. Michal Fec̆kan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. JinRong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JinRong Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fec̆kan, M., Wang, J. & Zhou, Y. Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators. J Optim Theory Appl 156, 79–95 (2013). https://doi.org/10.1007/s10957-012-0174-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0174-7

Keywords

Search

Navigation