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Approximate Controllability of Semilinear Fractional Control Systems of Order \({{\alpha \in (1, 2]}}\) with Infinite Delay

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Abstract

The objective of this paper is to present some sufficient conditions for approximate controllability of semilinear fractional control system of order \({\alpha \in (1, 2]}\) with infinite delay. The results are obtained by the theory of strongly continuous \({\alpha}\)-order cosine family and sequence method. At the end, an example is given to illustrate the theory.

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References

  1. Balachandran K., Dauer J.P.: Controllability of nonlinear systems in Banach spaces: a survey. J. Optim. Theory Appl. 115(1), 7–28 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balachandran K., Anthoni S.M.: Controllability of second-order semilinear neutral functional differential systems in Banach spaces. Comput. Math. Appl. 41(10-11), 1223–1235 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barnett S.: Introduction to Mathematical Control Theory. Clarendon Press, Oxford (1975)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  5. Henríquez H.R., Hernández E.M.: Approximate controllability of second-order distributed implicit functional systems. Nonlinear Anal. 70(2), 1023–1039 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jeong, J.M., Kwun, Y.C., Park, J.Y.: Approximate controllability of semilinear retarded functional differential equations. J. Dyn. Control Syst. 5, 329–346 (1999)

  7. Kalman, R.E.: Controllability of linear systems,contributions to differential equations. 1, 190–213 (1963)

  8. Kumar S., Sukavanam N.: Approximate controllability of fractional order semilinear systems with bounded delay. J. Differ. Equ. 252(11), 6163–6174 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kumar S., Sukavanam N.: On the approximate controllability of fractional order control systems with delay. Nonlinear Dyn. Syst. Theory 13(1), 69–78 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Kumar S., Sukavanam N.: Controllability of second-order systems with nonlocal conditions in Banach spaces. Numer. Funct. Anal. Optim. 35(4), 423–431 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, K., Peng, J., Gao, J.: Controllability of nonlocal fractional differential systems of order \({{\alpha \in (1, 2]}}\) in Banach spaces. Rep. Math. Phys. 71(1), 33 (2013)

    Article  MathSciNet  Google Scholar 

  12. Naito K.: Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim. 25, 715–722 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Naito K., Park J.Y.: Approximate controllability for trajectories of a delay volterra control system. J. Optim. Theory Appl. 61(2), 271–279 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pazy A.: Semigroup of Linear Operators and Application to Partial Differential Equations. Spinger, New York (1983)

    Book  MATH  Google Scholar 

  15. Sakthivel R., Ren Y., Mahmudov N.I.: On the approximate controllability of semilinear fractional differential systems. Comput. Math. Appl. 62(3), 1451–1459 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sakthivel R., Mahmudov N.I., Nieto J.J.: Controllability for a class of fractional-order neutral evolution control systems. Appl. Math. Comput. 218(20), 10334–10340 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Shukla, A., Arora, U., Sukavanam, N.: Approximate controllability of retarded semilinear stochastic system with non local conditions. J. Appl. Math. Comput. 49(1–2), 513–527 (2015)

  18. Sukavanam, N.: Approximate controllability of semilinear control systems with growing nonlinearity. In: Mathematical theory of control proceedings of international conference, Marcel Dekker, New York, pp 353–357 (1993).

  19. Sukavanam N., Tafesse S.: Approximate controllability of a delayed semilinear control system with growing nonlinear term. Nonlinear Anal. 74(18), 6868–6875 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Travis C.C., Webb G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hungar. 32(1-2), 75–96 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang J., Zhou Y.: Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal. Real World Appl. 12(6), 3642–3653 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang J., Zhou Y.: Analysis of nonlinear fractional control systems in Banach spaces. Nonlinear Anal. 74(17), 5929–5942 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhou H.X.: Approximate controllability for a class of semilinear abstract equations. SIAM J. Control Optim. 21(4), 551–565 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhou Y., Jiao F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59(3), 1063–1077 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific Publishing Co. Pte. Ltd, Hackensack, NJ (2014)

  26. Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Elsevier, New York (2015)

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    A. Shukla, N. Sukavanam & D. N. Pandey

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  1. A. Shukla
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  2. N. Sukavanam
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  3. D. N. Pandey
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Correspondence to A. Shukla.

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Shukla, A., Sukavanam, N. & Pandey, D.N. Approximate Controllability of Semilinear Fractional Control Systems of Order \({{\alpha \in (1, 2]}}\) with Infinite Delay. Mediterr. J. Math. 13, 2539–2550 (2016). https://doi.org/10.1007/s00009-015-0638-8

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  • DOI: https://doi.org/10.1007/s00009-015-0638-8

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