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Selected Problems of Controllability of Semilinear Fractional Systems-A Survey

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Intelligent Information and Database Systems (ACIIDS 2019)

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Abstract

The following article presents recent results of controllability problem of dynamical systems in infinite and finite-dimensional spaces. Roughly speaking, we describe selected controllability problems of fractional order systems, including approximate controllability and complete controllability.

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References

  1. Agarwal, R.P., de Andrade, B., Siracusa, G.: On fractional integro-differential equations with state-dependent delay. Comput. Math. Appl. 62(3), 1143–1149 (2011)

    MathSciNet  MATH  Google Scholar 

  2. de Andrade, B., Dos Santos, J.P.C.: Existence of solution for a fractional neutral integro-differential equation with unbounded delay. Electron. J. Differ. Equ. 2012(90), 1–13 (2012)

    MathSciNet  Google Scholar 

  3. Babiarz, A., Klamka, J., Niezabitowski, M.: Schauder’s fixed-point theorem in approximate controllability problems. Int. J. Appl. Math. Comput. Sci. 26(2), 263–275 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Bachelier, O., Dabkowski, P., Galkowski, K., Kummert, A.: Fractional and nd systems: a continuous case. Multidimension. Syst. Sig. Process. 23(3), 329–347 (2012)

    MATH  Google Scholar 

  5. Balasubramaniam, P., Tamilalagan, P.: Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi’s function. Appl. Math. Comput. 256, 232–246 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Balasubramaniam, P., Vembarasan, V., Senthilkumar, T.: Approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space. Numer. Funct. Anal. Optim. 35(2), 177–197 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Debbouche, A., Torres, D.F.M.: Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions. Fractional Calc. Appl. Anal. 18(1), 95–121 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Debbouche, A., Torres, D.F.: Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions. Appl. Math. Comput. 243, 161–175 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Fubini, G.: Sugli integrali multipli. Rendiconti Reale Accademia dei Lincei 5, 608–614 (1907)

    MATH  Google Scholar 

  11. Guechi, S., Debbouche, A., Torres, D.F.M.: Approximate controllability of impulsive non-local non-linear fractional dynamical systems and optimal control. Miskolc Math. Notes 19(1), 255–271 (2018)

    MathSciNet  Google Scholar 

  12. Guendouzi, T., Farahi, S.: Approximate controllability of Sobolev-type fractional functional stochastic integro-differential systems. Boletín de la Sociedad Matemática Mexicana 21(2), 289–308 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011)

    MATH  Google Scholar 

  14. Kaczorek, T., Sajewski, L.: The Realization Problem for Positive and Fractional Systems, vol. 1. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04834-5

    MATH  Google Scholar 

  15. Karthikeyan, S., Balachandran, K., Sathya, M.: Controllability of nonlinear stochastic systems with multiple time-varying delays in control. Int. J. Appl. Math. Comput. Sci. 25(2), 207–215 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam (2006). No. 13

    MATH  Google Scholar 

  17. Klamka, J., Babiarz, A., Niezabitowski, M.: Banach fixed-point theorem in semilinear controllability problems-a survey. Bull. Pol. Acad. Sci. Tech. Sci. 64(1), 21–35 (2016)

    MATH  Google Scholar 

  18. Klamka, J.: Schauder’s fixed-point theorem in nonlinear controllability problems. Control Cybern. 29, 153–165 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Kumlin, P.: Functional Analysis Notes: A Note on Fixed Point Theory. Chalmers & GU, Gothenburg (2004)

    Google Scholar 

  20. Liang, J., Yang, H.: Controllability of fractional integro-differential evolution equations with nonlocal conditions. Appl. Math. Comput. 254, 20–29 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Liang, S., Mei, R.: Existence of mild solutions for fractional impulsive neutral evolution equations with nonlocal conditions. Adv. Differ. Equ. 2014(1), 101 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Liu, Z., Li, X.: On the controllability of impulsive fractional evolution inclusions in Banach spaces. J. Optim. Theory Appl. 156(1), 167–182 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Liu, Z., Li, X.: On the exact controllability of impulsive fractional semilinear functional differential inclusions. Asian J. Control 17(5), 1857–1865 (2015)

    MathSciNet  MATH  Google Scholar 

  24. Mur, T., Henriquez, H.R.: Relative controllability of linear systems of fractional order with delay. Math. Control Relat. Fields 5(4), 845–858 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Nirmala, R.J., Balachandran, K.: The controllability of nonlinear implicit fractional delay dynamical systems. Int. J. Appl. Math. Comput. Sci. 27(3), 501–513 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Paszke, W., Dabkowski, P., Rogers, E., Gałkowski, K.: New results on strong practical stability and stabilization of discrete linear repetitive processes. Syst. Control Lett. 77, 22–29 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Schmeidel, E., Zbaszyniak, Z.: An application of Darbo’s fixed point theorem in the investigation of periodicity of solutions of difference equations. Comput. Math. Appl. 64(7), 2185–2191 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Wang, J., Fan, Z., Zhou, Y.: Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. J. Optim. Theory Appl. 154(1), 292–302 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Wang, J., Zhou, Y.: Complete controllability of fractional evolution systems. Commun. Nonlinear Sci. Numer. Simul. 17(11), 4346–4355 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Xiaoli, D., Nieto, J.J.: Controllability and optimality of linear time-invariant neutral control systems with different fractional orders. Acta Math. Sci. 35(5), 1003–1013 (2015)

    MathSciNet  MATH  Google Scholar 

  31. Yan, Z.: Existence results for fractional functional integrodifferential equations with nonlocal conditions in Banach spaces. Annales Polonici Mathematici 3, 285–299 (2010)

    MathSciNet  MATH  Google Scholar 

  32. Yan, Z.: Controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay. J. Franklin Inst. 348(8), 2156–2173 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Yan, Z.: Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay. Int. J. Control 85(8), 1051–1062 (2012)

    MathSciNet  MATH  Google Scholar 

  34. Yang, H., Ibrahim, E.: Approximate controllability of fractional nonlocal evolution equations with multiple delays. Adv. Differ. Equ. 2017(1), 272 (2017)

    MathSciNet  MATH  Google Scholar 

  35. Zawiski, R.: On controllability and measures of noncompactness. In: AIP Conference Proceedings, vol. 1637, pp. 1241–1246. AIP (2014)

    Google Scholar 

  36. Zhang, X., Huang, X., Liu, Z.: The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay. Nonlinear Anal.: Hybrid Syst. 4(4), 775–781 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Zhang, X., Zhu, C., Yuan, C.: Approximate controllability of fractional impulsive evolution systems involving nonlocal initial conditions. Adv. Differ. Equ. 2015(1), 244 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Zhou, X.F., Wei, J., Hu, L.G.: Controllability of a fractional linear time-invariant neutral dynamical system. Appl. Math. Lett. 26(4), 418–424 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Elsevier Science, Amsterdam (2016)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  41. Zhou, Y., Vijayakumar, V., Murugesu, R.: Controllability for fractional evolution inclusions without compactness. Evol. Equ. Control Theory 4(4), 507–524 (2015)

    MathSciNet  MATH  Google Scholar 

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Acknowledgment

The research presented here was done by authors as parts of the projects funded by the National Science Centre in Poland granted according to decision UMO-2017/27/B/ST6/00145 (JK) and DEC-2015/19/D/ST7/03679 (AB).

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Authors and Affiliations

  1. Silesian University of Technology, Akademicka 2A, 44-100, Gliwice, Poland

    Artur Babiarz

  2. Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100, Gliwice, Poland

    Jerzy Klamka

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  1. Artur Babiarz
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Correspondence to Artur Babiarz .

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Editors and Affiliations

  1. Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Ngoc Thanh Nguyen

  2. Bina Nusantara University, Jakarta, Indonesia

    Ford Lumban Gaol

  3. National University of Kaohsiung, Kaohsiung, Taiwan

    Tzung-Pei Hong

  4. Wrocław University of Science and Technology, Wrocław, Poland

    Bogdan Trawiński

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Babiarz, A., Klamka, J. (2019). Selected Problems of Controllability of Semilinear Fractional Systems-A Survey. In: Nguyen, N., Gaol, F., Hong, TP., Trawiński, B. (eds) Intelligent Information and Database Systems. ACIIDS 2019. Lecture Notes in Computer Science(), vol 11431. Springer, Cham. https://doi.org/10.1007/978-3-030-14799-0_34

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  • DOI: https://doi.org/10.1007/978-3-030-14799-0_34

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