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Nonlocal Controllability of Semilinear Dynamic Systems with Fractional Derivative in Banach Spaces

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Abstract

In this paper, we establish two sufficient conditions for nonlocal controllability for fractional evolution systems. Since there is no compactness of characteristic solution operators, our theorems guarantee the effectiveness of controllability results under some weakly compactness conditions.

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References

  1. Byszewski, L.: Theorems about existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Fan, Z., Li, G.: Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct. Anal. 258, 1709–1727 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hernández, E., Santos, J.S., Azevedo, K.A.G.: Existence of solutions for a class of abstract differential equations with nonlocal conditions. Nonlinear Anal. 74, 2624–2634 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Wang, J., Wei, W.: A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces. Results Math. 58, 379–397 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hernández, E., O’Regan, D.: Controllability of Volterra–Fredholm type systems in Banach space. J. Franklin Inst. 346, 95–101 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

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

    Article  MATH  Google Scholar 

  8. 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 

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

    MathSciNet  MATH  Google Scholar 

  10. 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 

  11. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Jaradat, O.K., Al-Omari, A., Momani, S.: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear Anal. 69, 3153–3159 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Banas̀, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure and Appl. Math., vol. 60. Marcel Dekker, New York (1980)

    Google Scholar 

  15. Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. de Gruyter Ser. Nonlinear Anal. Appl., vol. 7. de Gruyter, Berlin (2001)

    Book  MATH  Google Scholar 

  16. 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 

  17. O’Regan, D., Precup, R.: Existence criteria for integral equations in Banach spaces. J. Inequal. Appl. 6, 77–97 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Mönch, H.: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the referees for careful reading of the manuscript and insightful comments, which helped to improve the quality of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improve the presentation of the paper. Finally, the first author acknowledges the support by the Key Projects of Science Technology Research in the Chinese Ministry of Education (211169), Tianyuan Special Funds of the National Natural Science Foundation of China (11026102); the second author acknowledges the support by the National Natural Science Foundation of China (11001034); the third author acknowledges the support by the National Natural Science Foundation of China (10971173).

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Authors and Affiliations

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

    JinRong Wang

  2. Department of Mathematics, Changshu Institute of Technology, Suzhou, 215500, P.R. China

    Zhenbin Fan

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

    Yong Zhou

Authors
  1. JinRong Wang
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  2. Zhenbin Fan
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  3. Yong Zhou
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Corresponding author

Correspondence to JinRong Wang.

Additional information

Communicated by Mark J. Balas.

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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). https://doi.org/10.1007/s10957-012-9999-3

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  • DOI: https://doi.org/10.1007/s10957-012-9999-3

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