Advertisement

Journal of Dynamical and Control Systems

, Volume 25, Issue 2, pp 245–262 | Cite as

Approximate Controllability for Nonlocal Fractional Propagation Systems of Sobolev Type

  • Xianghu Liu
  • JinRong WangEmail author
  • Yong Zhou
Article
  • 176 Downloads

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.

Keywords

Fractional propagation systems Sobolev type Nonlocal conditions Compactness of solution set Approximate controllability 

Mathematics Subject Classification (2010)

47J35 93B05 93C25 

Notes

Funding Information

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).

References

  1. 1.
    Miller KS, Ross B. An introduction to the fractional calculus and differential equations. New York: John Wiley; 1993.zbMATHGoogle Scholar
  2. 2.
    Podlubny I. Fractional differential equations. San Diego: Academic Press; 1999.zbMATHGoogle Scholar
  3. 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.Google Scholar
  4. 4.
    El-Borai MM. Some probability densities and fundamental solutions of fractional evolution equations. Chaos Sol Frac 2002;14:433–440.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Zhou Y, Jiao F. Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal 2010;11:4465–4475.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kumar S, Sukavanam N. Approximate controllability of fractional order semilinear systems with bounded delay. J Diff Equ 2012;252:6163–6174.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Aissani K, Benchohra M. Controllability of fractional integrodifferential equations with state-dependent delay. J Integr Equ Appl 2016;28:149–167.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liu Z, Li X. Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives. SIAM J Control Optim 2015;53:1920–1933.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hamani S, Henderson J. Boundary value problems for fractional differential inclusions with nonlocal conditions. Mediterr J Math 2016;13:967–979.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Zhou Y, Vijayakumar V, Murugesu R. Controllability for fractional evolution inclusions without compactness. Evol Equ Control The 2015;4:507–524.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Yang M, Wang QR. Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. Math Meth Appl Sci 2017;40:1126–1138.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Debbouche A, Baleanu D. Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput Math Appl 2011;62:1442–1450.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Debbouche A, Torres DFM. Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces. Int J Control 2013;86:1577–1585.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lightbourne JH, Rankin SM. A partial functional differential equation of Sobolev type. J Math Anal Appl 1983;93:328–337.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ponce R. Hölder continuous solutions for Sobolev type differential equations. Math Nachr 2014;287:70–78.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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.zbMATHGoogle Scholar
  22. 22.
    Fan Z, Dong Q, Li G. Approximate controllability for semilinear composite fractional relaxation equations. Fract Calc Appl Anal 2016;19:267–284.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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.MathSciNetzbMATHGoogle Scholar
  24. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fu X, Rong H. Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions. Autom Remote Control 2016;77:428–442.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 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.CrossRefzbMATHGoogle Scholar
  29. 29.
    Wang J, Fečkan M, Zhou Y. Controllability of Sobolev type fractional evolution systems. Dyn Part Differ Equ 2014;11:71–87.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Liu Z, Zeng B. Existence and controllability for fractional evolution inclusions of Clarke’s subdifferential type. Appl Math Comput 2015;257:178–189.MathSciNetzbMATHGoogle Scholar
  31. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Liang J, Xiao TJ. Abstract degenerate Cauchy problems in locally convex spaces. J Math Anal Appl 2001;259:398–412.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Akhmerov RR, Kamenskii M, Potapov AS, Rodkina AE, Sadovskii BN. Measures of noncompactness and condensing operators. Boston: Birkhäser; 1992.CrossRefzbMATHGoogle Scholar
  35. 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.CrossRefzbMATHGoogle Scholar
  36. 36.
    Pazy A. Semigroups of linear operators and applications to partial differential equations. Berlin: Springer; 1983.CrossRefzbMATHGoogle Scholar
  37. 37.
    Wang J, Li X. A uniform method to Ulam-Hyers stability for some linear fractional equations. Mediterr J Math 2016;13:625–635.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Curtain RF, Zwart H. An introduction to infinite dimensional linear systems theory. New York: Springer-Verlag; 1995.CrossRefzbMATHGoogle Scholar
  39. 39.
    Mahmudov NI. Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J Control Optim 2003;42:1604–1622.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsGuizhou UniversityGuiyangPeople’s Republic of China
  2. 2.Department of MathematicsZunyi Normal CollegeZunyiPeople’s Republic of China
  3. 3.Department of MathematicsXiangtan UniversityXiangtanPeople’s Republic of China
  4. 4.Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations