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Boundary Controllability of Riemann–Liouville Fractional Semilinear Evolution Systems

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Abstract

The main objective of this work is to steer a semilinear time-fractional diffusion control system involving Riemann–Liouville fractional derivative to a desired state in a part of the boundary of the evolution domain. For that, we use fixed point technique, semigroup theory and fractional calculus under some proposed assumptions in the linear part of the system and the nonlinear term. At the end, we provide some numerical simulations which lead to successful figures, in order to guarantee the efficiency of the proposed approach.

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References

  1. Axtell, M., Bise, M.E.: Fractional Calculus Applications in Control Systems, presented at the 1990 National Aerospace and Electronics Conference. Dayton, OH, USA (1990)

  2. Baleanu, D., Fahimeh, A.G., Juan, J.N., Jajarmi, A.: On a new and generalized fractional model for a real cholera outbreak. Alex. Eng. J. 61(11), 9175–9186 (2022)

    Article  Google Scholar 

  3. Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics. Springer, New York (2010)

    Book  MATH  Google Scholar 

  4. Ebenezer, B., Paul, A., Canan, U.: Mathematical modeling of transmission of water pollution. J. Prime Res. Math. 17(2), 20–38 (2021)

    MathSciNet  Google Scholar 

  5. Fudong, G., Chen, Y.Q., Kou, C.: Regional Analysis of Time-Fractional Diffusion Processes. Springer, New York (2018)

    MATH  Google Scholar 

  6. Fudong, G., YangQuan, C., Chunhai, K., Podlubny, R.: On the regional controllability of the sub-diffusion process with Caputo fractional derivative. Fract. Calc. Appl. Anal. 19(5), 1262–1281 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gutiérrez, R.E., Rosário, J.M., Machado, J.T.: Fractional order calculus: basic concepts and engineering applications. Math. Probl. Eng. 2010, 1–19 (2010)

    Article  MATH  Google Scholar 

  8. Hamdy, M.A., El-Borai, M.M., Ramadan, M.. El..: Boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and Poisson jumps. Adv. Differ. Equ. 1, 1–23 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Hamdy, M.A.: Conformable fractional stochastic differential equations with control function. Syst. Control Lett. 158, 105062 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hamdy, M.A., El-Borai, M.M., El-Owaidy, H.M., Ghanem, A.S.: Existence solution and controllability of Sobolev type delay nonlinear fractional integro-differential system. Mathematics 7(1), 79p (2019)

    Article  MathSciNet  Google Scholar 

  11. Hamdy, M.A.: Boundary controllability of nonlinear fractional integrodifferential systems. Adv. Differ. Equ. 2010, 1–9 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Hamdy, M.A., JinRong, W.: Exact null controllability of Sobolev-type Hilfer fractional stochastic differential equations with fractional Brownian motion and Poisson jumps. Bull. Iran. Math. Bull. 44(3), 673–690 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheol. Acta 45, 765–771 (2006)

    Article  Google Scholar 

  14. Hoang, D.T.: Controllability and Observability of non Autonomous Evolution Equations. Optimization and control, Université de Bordeaux (2018)

  15. Jinrong, W., Hamdy, M.A.: Null controllability of nonlocal Hilfer fractional stochastic differential equations. Miskolc Math. Notes 18(2), 1073–1083 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Karite, T., Boutoulout, A.: Regional boundary controllability of semilinear parabolic systems with state constraints. Int. J. Dyn. Syst. Differ. Equ. 8(1/2), 150–159 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Kavitha, K., Vijayakumar, V., Udhayakumar, R.: Results on controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncompactness. Chaos Solitons Fractals 139, 110035 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kavitha, K., Vijayakumar, V., Shukla, A., Nisar, K.S., Udhayakumar, R.: Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type. Chaos Solitons Fractals 151, 111264 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  20. Kou, S.: Stochastic modeling in nanoscale biophysics: sub-diffusion within proteins. Ann. Appl. Stat. 2, 501–538 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, X., Liu, Z., Bin, M.: Approximate controllability of impulsive fractional neutral evolution equations with Riemann–Liouville fractional derivatives. J. Comput. Anal. Appl. 17(3), 468–485 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Mahmudov, N.I., Zorlu, S.: On the approximate controllability of fractional evolution equations with compact analytic semigroup. J. Comput. Appl. Math. 259, 194–204 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mandelbrot, B.B.: The Fractal Geometry of Nature. Macmillan, New York (1983)

    Book  Google Scholar 

  24. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  25. Nigmatulin, R.R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. B 133, 425–430 (1986)

    Article  Google Scholar 

  26. Oldham, K.B., Spanier, J.: Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

  27. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (2012)

    MATH  Google Scholar 

  28. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  29. Tajani, A., Alaoui, F.-Z El., Boutoulout, A.: Regional boundary controllability of semilinear subdiffusion Caputo fractional systems. Math. Comput. Simul. 193, 481–496 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tajani, A., ElAlaoui, F.Z., Boutoulout, A.: Regional controllability of a class of time-fractional systems. In: Hammouch, Z., Dutta, H., Melliani, S., Ruzhansky, M. (eds.) Nonlinear Analysis: Problems, Applications and Computational Methods. SM2A 2019. Lecture Notes in Networks and Systems, vol. 168, pp. 141–155. Springer, New York (2021)

    Google Scholar 

  31. Tajani, A., El Alaoui, F.Z., Boutoulout, A.: Regional controllability of Riemann–Liouville time-fractional semilinear evolution equations. Math. Probl. Eng. 2020, 1–7 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294–298 (1984)

    Article  MATH  Google Scholar 

  33. Tusset, Angelo M., Danilo, I., Maria, E.K.F., Priscilla, M.L.Z.C., Giane, G.L.: Dynamic analysis and control for a bioreactor in fractional order. Symmetry 14(8), 1609 (2022)

    Article  Google Scholar 

  34. Uchaikin, V., Sibatov, R.: Fractional Kinetics in Solids: Anomalous Charge Transport in Semiconductors. World Scientific Publishing, Singapore (2013)

    Book  MATH  Google Scholar 

  35. Wang, J., Fe, K.M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8(4), 345–361 (2011)

    Article  MathSciNet  Google Scholar 

  36. Wang, R.-N., Chen, D.-H., Xiao, T.-J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252(1), 202–235 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Zerrik, E., El Jai, A., Boutoulout, A.: Actuators and regional boundary controllability of parabolic system. Int. J. Syst. Sci 31(1), 73–82 (2000)

    Article  MATH  Google Scholar 

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Acknowledgements

This paper is in memory of the late professor Ali Boutoulout from Moulay Ismail University—Faculty of Sciences in Meknes—Morocco, who contributed to the realization of this work.

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Authors and Affiliations

  1. TSI Team, Moulay Ismail University, Faculty of Sciences, 11201, Zitoune, Meknes, Morocco

    Asmae Tajani & Fatima-Zahrae El Alaoui

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  1. Asmae Tajani
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  2. Fatima-Zahrae El Alaoui
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Correspondence to Asmae Tajani.

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Communicated by Hoang Xuan Phu.

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Tajani, A., El Alaoui, FZ. Boundary Controllability of Riemann–Liouville Fractional Semilinear Evolution Systems. J Optim Theory Appl 198, 767–780 (2023). https://doi.org/10.1007/s10957-023-02248-7

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  • DOI: https://doi.org/10.1007/s10957-023-02248-7

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