Skip to main content
SpringerLink
Log in

Approximate Controllability for Nonlocal Fractional Propagation Systems of Sobolev Type

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

This paper is concerned with the approximate controllability of a class of fractional propagation systems of Sobolev type with nonlocal conditions in Hilbert spaces. By utilizing the theory of propagation family and techniques of measures of noncompactness, we firstly prove existence of the mild solutions and compactness of solutions set in Banach spaces under weaker conditions, i.e., the compactness condition of propagation family is instead of norm continuous in the sense of uniform operator topology. Secondly, we establish the principle for linear fractional propagation systems is approximately controllable in Hilbert spaces. Then, we present an interesting criteria for approximate controllability of semilinear fractional propagation systems in Hilbert spaces by requiring the corresponding linear fractional propagation systems is approximately controllable and assuming that the resolvent set of the operator pair is compact. Finally, a fractional PDEs model is used to demonstrate that how to check the conditions in the above abstract theorems, In particular, a principle for approximate controllability of linear problem is verified by using the property of Mittag-Leffler function.

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. Miller KS, Ross B. An introduction to the fractional calculus and differential equations. New York: John Wiley; 1993.

    MATH  Google Scholar 

  2. Podlubny I. Fractional differential equations. San Diego: Academic Press; 1999.

    MATH  Google Scholar 

  3. Kilbas AA, Srivastava HM, Trujillo J. Theory and applications of fractional differential equations. North-holland mathematics studies. Amsterdam: Elsevier Science B.V.; 2006. p. 204.

  4. El-Borai MM. Some probability densities and fundamental solutions of fractional evolution equations. Chaos Sol Frac 2002;14:433–440.

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhou Y, Jiao F. Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal 2010;11:4465–4475.

    Article  MathSciNet  MATH  Google Scholar 

  6. Kumar S, Sukavanam N. Approximate controllability of fractional order semilinear systems with bounded delay. J Diff Equ 2012;252:6163–6174.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Aissani K, Benchohra M. Controllability of fractional integrodifferential equations with state-dependent delay. J Integr Equ Appl 2016;28:149–167.

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu Z, Li X. Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives. SIAM J Control Optim 2015;53:1920–1933.

    Article  MathSciNet  MATH  Google Scholar 

  10. Mahmudov NI, Zorlu S. On the approximate controllability of fractional evolution equations with compact analytic semigroup. J Comput Appl Math 2014;259:194–204.

    Article  MathSciNet  MATH  Google Scholar 

  11. Hamani S, Henderson J. Boundary value problems for fractional differential inclusions with nonlocal conditions. Mediterr J Math 2016;13:967–979.

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhou Y, Vijayakumar V, Murugesu R. Controllability for fractional evolution inclusions without compactness. Evol Equ Control The 2015;4:507–524.

    Article  MathSciNet  MATH  Google Scholar 

  13. Hernández E, O’Regan D, Balachandran K. Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators. Indagat Math 2013;24:68–82.

    Article  MathSciNet  MATH  Google Scholar 

  14. Yang M, Wang QR. Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. Math Meth Appl Sci 2017;40:1126–1138.

    Article  MathSciNet  MATH  Google Scholar 

  15. Debbouche A, Baleanu D. Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput Math Appl 2011;62:1442–1450.

    Article  MathSciNet  MATH  Google Scholar 

  16. Debbouche A, Torres DFM. Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces. Int J Control 2013;86:1577–1585.

    Article  MathSciNet  MATH  Google Scholar 

  17. Guendouzi T, Farahi S. Approximate controllability of semilinear fractional stochastic dynamic systems with nonlocal conditions in Hilbert spaces. Mediterr J Math 2016;13:637–656.

    Article  MathSciNet  MATH  Google Scholar 

  18. Mokkedem FZ, Fu X. Approximate controllability for a semilinear stochastic evolution system with infinite delay in l p space. Appl Math Optim 2017;75:253–283.

    Article  MathSciNet  MATH  Google Scholar 

  19. Lightbourne JH, Rankin SM. A partial functional differential equation of Sobolev type. J Math Anal Appl 1983;93:328–337.

    Article  MathSciNet  MATH  Google Scholar 

  20. Ponce R. Hölder continuous solutions for Sobolev type differential equations. Math Nachr 2014;287:70–78.

    Article  MathSciNet  MATH  Google Scholar 

  21. Kerboua M, Debbouche A, Baleanu D. Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces, Electron. J Qual Theory Differ Equ 2014;58:1–16.

    MATH  Google Scholar 

  22. Fan Z, Dong Q, Li G. Approximate controllability for semilinear composite fractional relaxation equations. Fract Calc Appl Anal 2016;19:267–284.

    Article  MathSciNet  MATH  Google Scholar 

  23. Debbouche A, Nieto JJ. Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls. Appl Math Comput 2014;245:74–85.

    MathSciNet  MATH  Google Scholar 

  24. Debbouche A, Torres DFM. Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions. Fract Calc Appl Anal 2015;18:95–121.

    Article  MathSciNet  MATH  Google Scholar 

  25. Fečkan M, Wang J, Zhou Y. Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators. J Optim Theory Appl 2013;156:79–95.

    Article  MathSciNet  MATH  Google Scholar 

  26. Fu X, Rong H. Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions. Autom Remote Control 2016;77:428–442.

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang J, Fečkan M, Zhou Y. Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions. Evol Equ Control The 2017; 6:471–486.

    Article  MathSciNet  MATH  Google Scholar 

  28. Li F, Liang J, Xu HK. Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions. J Math Anal Appl 2012;391:510–525.

    Article  MATH  Google Scholar 

  29. Wang J, Fečkan M, Zhou Y. Controllability of Sobolev type fractional evolution systems. Dyn Part Differ Equ 2014;11:71–87.

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu Z, Zeng B. Existence and controllability for fractional evolution inclusions of Clarke’s subdifferential type. Appl Math Comput 2015;257:178–189.

    MathSciNet  MATH  Google Scholar 

  31. Aghajani A, Pourhadi E, Trujillo J. Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces. Fract Calc Appl Anal 2013;16:962–977.

    Article  MathSciNet  MATH  Google Scholar 

  32. Jiang Y, Huang N, Yao J. Solvability and optimal control of semilinear nonlocal fractional evolution inclusion with Clarke subdifferential. Appl Anal 2017;96:2349–2366.

    Article  MathSciNet  MATH  Google Scholar 

  33. Liang J, Xiao TJ. Abstract degenerate Cauchy problems in locally convex spaces. J Math Anal Appl 2001;259:398–412.

    Article  MathSciNet  MATH  Google Scholar 

  34. Akhmerov RR, Kamenskii M, Potapov AS, Rodkina AE, Sadovskii BN. Measures of noncompactness and condensing operators. Boston: Birkhäser; 1992.

    Book  MATH  Google Scholar 

  35. Kamenskii M, Obukhovskii V, Zecca P, Vol. 7. Condensing multivalued maps and semilinear differential inclusions in Banach spaces, de Gruyter Ser. Nonlinear Anal. Appl. Berlin: Walter de Gruyter; 2001.

    Book  MATH  Google Scholar 

  36. Pazy A. Semigroups of linear operators and applications to partial differential equations. Berlin: Springer; 1983.

    Book  MATH  Google Scholar 

  37. Wang J, Li X. A uniform method to Ulam-Hyers stability for some linear fractional equations. Mediterr J Math 2016;13:625–635.

    Article  MathSciNet  MATH  Google Scholar 

  38. Curtain RF, Zwart H. An introduction to infinite dimensional linear systems theory. New York: Springer-Verlag; 1995.

    Book  MATH  Google Scholar 

  39. Mahmudov NI. Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J Control Optim 2003;42:1604–1622.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work is partially supported by the National Natural Science Foundation of China (11661016, 11671339); Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006); Science and Technology Program of Guizhou Province ([2017]5788); and Guizhou Province Science and Technology Fund ([2016]1160).

Author information

Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, People’s Republic of China

    Xianghu Liu & JinRong Wang

  2. Department of Mathematics, Zunyi Normal College, Zunyi, 563006, Guizhou, People’s Republic of China

    Xianghu Liu

  3. Department of Mathematics, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Yong Zhou

  4. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Yong Zhou

Authors
  1. Xianghu Liu
    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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Wang, J. & Zhou, Y. Approximate Controllability for Nonlocal Fractional Propagation Systems of Sobolev Type. J Dyn Control Syst 25, 245–262 (2019). https://doi.org/10.1007/s10883-018-9409-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-018-9409-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation