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Controllability Results for Non Densely Defined Impulsive Fractional Differential Equations in Abstract Space

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Abstract

In this paper, we study controllability results for non-densely defined impulsive fractional differential equation by applying the concepts of semigroup theory, fractional calculus, and Banach Fixed Point Theorem. An example is also discussed to illustrate the obtained results.

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References

  1. Abbas, S.: Existence of solutions to fractional order ordinary and delay differential equations and applications. Electronic Journal of Differential Equations (EJDE)[electronic only], 2011:Paper–No (2011)

  2. Abbas, S., Bahuguna,D.: Existence of solutions to quasilinear functional differential equations. Electronic Journal of Differential Equations (EJDE) [electronic only]:Paper No (2009)

  3. Adimy, M., Laklach, M., Ezzinbi, K.: Non-linear semigroup of a class of abstract semilinear functional differential equations with a non-dense domain. Acta Mathematica Sinica 20(5), 933–942 (2004)

    Article  MathSciNet  Google Scholar 

  4. Balachandran, K., Govindaraj, V., Rodriguez-Germa, L., Trujillo, J.: Controllability of nonlinear higher order fractional dynamical systems. Nonlinear Dyn. 71(4), 605–612 (2013)

    Article  MathSciNet  Google Scholar 

  5. Balasubramaniam, P., Park, J., Muthukumar, P.: Approximate controllability of neutral stochastic functional differential systems with infinite delay. Stoch. Anal. Appl. 28(2), 389–400 (2010)

    Article  MathSciNet  Google Scholar 

  6. Chadha, A., Bora, S., Sakthivel, R.: Approximate controllability of impulsive stochastic fractional differential equations with nonlocal conditions. Dyn. Syst. Appl. 27(1), 1–29 (2018)

    Google Scholar 

  7. Da Prato, G., Sinestrari, E.: Differential operators with non dense domain. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 14(2), 285–344 (1987)

    MathSciNet  MATH  Google Scholar 

  8. Dauer, J., Mahmudov, N.: Controllability of some nonlinear systems in hilbert spaces. J. Optim. Theory Appl. 123(2), 319–329 (2004)

    Article  MathSciNet  Google Scholar 

  9. Feckan, M., Zhou, Y., Wang, J.: Response to comments on the concept of existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 19, 401–3 (2014)

    Article  MathSciNet  Google Scholar 

  10. Fu, X.: Controllability of non-densely defined functional differential systems in abstract space. Appl. Math. Lett. 19(4), 369–377 (2006)

    Article  MathSciNet  Google Scholar 

  11. Hernandez, E., Sakthivel, R., Aki, S.T.: Existence results for impulsive evolution differential equations with state-dependent delay. Electronic Journal of Differential Equations (EJDE) [electronic only] 8:Paper No (2008)

  12. Hilal,K., Allaoui, Y.: Existence of solution for fractional impulsive differential equations with non-dense domain. Available at SSRN 3271390 (2018)

  13. Kavitha, V., Arjunan, M.M.: Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in banach spaces. Nonlinear Anal.: Hybrid Syst. 4(3), 441–450 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Kellerman, H., Hieber, M.: Integrated semigroups. J. Funct. Anal. 84(1), 160–180 (1989)

    Article  MathSciNet  Google Scholar 

  15. Kilbas, A.A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations, 204th edn. Elsevier Science Limited, Amsterdam (2006)

    MATH  Google Scholar 

  16. Klamka, J.: Controllability of linear dynamical systems (1963)

  17. Kumar, A., Muslim, M., Sakthivel, R.: Controllability of the second-order nonlinear differential equations with non-instantaneous impulses. J. Dyn. Control Syst. 24(2), 325–342 (2018)

    Article  MathSciNet  Google Scholar 

  18. Lakshmikantham, V., Simeonov, P.S., et al.: Theory of impulsive differential equations, 6th edn. World scientific, Singapore (1989)

    Book  Google Scholar 

  19. Liu, J., Zhang, L.: Existence of anti-periodic (differentiable) mild solutions to semilinear differential equations with nondense domain. SpringerPlus 5(1), 704 (2016)

    Article  Google Scholar 

  20. Magal, P., Ruan, S., et al.: On semilinear cauchy problems with non-dense domain. Adv. Differ. Equ. 14(11/12), 1041–1084 (2009)

    MathSciNet  MATH  Google Scholar 

  21. Mahmudov, N.I.: Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J Control Optim. 42(5), 1604–1622 (2003)

    Article  MathSciNet  Google Scholar 

  22. Mahto, L., Abbas. S.: Approximate controllability and optimal control of impulsive fractional semilinear delay differential equations with non-local conditions. arXiv preprint arXiv:1304.3198 (2013)

  23. Mainardi, F., Paradisi, P., Gorenflo, R.: Probability distributions generated by fractional diffusion equations. arXiv preprint arXiv:0704.0320 (2007)

  24. Malik, M., Dhayal, R., Abbas, S., Kumar, A.: Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, pp. 1–16 (2017)

  25. Naito, K.: Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim. 25(3), 715–722 (1987)

    Article  MathSciNet  Google Scholar 

  26. Podlubny, I.: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Math. Sci. Eng. 198, 114–144 (1999)

    MATH  Google Scholar 

  27. Quinn, M., Carmichael, N.: An approach to non-linear control problems using fixed-point methods, degree theory and pseudo-inverses. Numer. Funct. Anal. Optim. 7(2–3), 197–219 (1985)

    Article  Google Scholar 

  28. Sakthivel, R., Nieto, J.J., Mahmudov, N., et al.: Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay. Taiwan. J. Math. 14(5), 1777–1797 (2010)

    Article  MathSciNet  Google Scholar 

  29. Sakthivel, R., Ren, Y., Debbouche, A., Mahmudov, N.: Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions. Appl. Anal. 95(11), 2361–2382 (2016)

    Article  MathSciNet  Google Scholar 

  30. Sakthivel, R., Revathi, P., Ren, Y.: Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal.: Theory, Methods Appl. 81, 70–86 (2013)

    Article  MathSciNet  Google Scholar 

  31. Sakthivel, R., Suganya, S., Anthoni, S.M.: Approximate controllability of fractional stochastic evolution equations. Comput. Math. Appl. 63(3), 660–668 (2012)

    Article  MathSciNet  Google Scholar 

  32. Simeonov, P.: Impulsive differential equations: periodic solutions and applications. Routledge, Abingdon (2017)

    Google Scholar 

  33. Subashini, R., Kumar, S.V., Saranya, S., Ravichandran, C.: On the controllability of non-densely defined fractional neutral functional differential equations in banach spaces. Int. J. Pure Appl. Math. 118(11), 257–276 (2018)

    Google Scholar 

  34. Tomar, N.K., Sukavanam, N.: Approximate controllability of non-densely defined semilinear delayed control systems. Nonlinear Stud. 18(2), 229–234 (2011)

    MathSciNet  MATH  Google Scholar 

  35. Vijayakumar, V.: Approximate controllability results for non-densely defined fractional neutral differential inclusions with hille–yosida operators. International Journal of Control, pp. 1–13 (2018)

  36. Vijayakumar, V., Ravichandran, C., Murugesu, R.: Approximate controllability for a class of fractional neutral integro-differential inclusions with state-dependent delay. Nonlinear studies 20(4), (2013)

  37. 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  Google Scholar 

  38. Wang, J., Zhou, Y., Fec, M., et al.: On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Comput. Math. Appl. 64(10), 3008–3020 (2012)

    Article  MathSciNet  Google Scholar 

  39. Z. Zhang and B. Liu. A note on impulsive fractional evolution equations with nondense domain. In Abstract and Applied Analysis, volume 2012. Hindawi (2012)

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Funding

The first author is thankful to the University Grant Commission for its financial support to carry out his research work.

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

    Ashish Kumar & Dwijendra N. Pandey

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  1. Ashish Kumar
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  2. Dwijendra N. Pandey
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Correspondence to Ashish Kumar.

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Kumar, A., Pandey, D.N. Controllability Results for Non Densely Defined Impulsive Fractional Differential Equations in Abstract Space. Differ Equ Dyn Syst 29, 227–237 (2021). https://doi.org/10.1007/s12591-019-00471-1

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