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Controllability of Prabhakar Fractional Dynamical Systems

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Abstract

In this manuscript, we focus on the investigation of controllability results in various types of Prabhakar fractional dynamical systems (FDSs). We establish sufficient and necessary conditions for the controllability of linear Prabhakar FDS. Additionally, we explore sufficient conditions for the controllability of nonlinear Prabhakar FDS, nonlinear integro-differential Prabhakar FDS, and nonlinear neutral Prabhakar FDS using the Schauder fixed point theorem. For the controllability of nonlinear Prabhakar PDSs, we assume that the nonlinear function is continuous instead of Lipschitz continuous. The solution of the neutral Prabhakar FDS is determined by applying the Laplace transform. Finally, we provide illustrative examples to validate and demonstrate the applicability of the obtained results.

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References

  1. Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)

    ADS  Google Scholar 

  2. Ahmed, E., Elgazzar, A.: On fractional order differential equations model for nonlocal epidemics. Physica A 379(2), 607–614 (2007)

    ADS  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  3. Acay, B., Ozarslan, R., Bas, E.: Fractional physical models based on falling body problem. AIMS Math. 5(3), 2608 (2020)

    MathSciNet  Google Scholar 

  4. Hilfer, R.: Applications of fractional calculus in physics (2000)

  5. Ahmed, H.M., El-Borai, M.M., El-Sayed, W.G., Elbadrawi, A.Y.: Fractional stochastic evolution inclusions with control on the boundary. Symmetry 15(4), 928 (2023)

    ADS  Google Scholar 

  6. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations 204. Elsevier (2006)

  7. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier (1998)

  8. Kai, D.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (2010)

  9. Ahmed, H.M., El-Borai, M.M., Ramadan, M.E.: Boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and poisson jumps. Adv. Differ. Equ. 2019(1), 1–23 (2019)

    MathSciNet  Google Scholar 

  10. Wang, S., Liu, J., Zhang, X.: Properties of solutions for fractional-order linear system with differential equations. AIMS Math. 7(8), 15704–15713 (2022)

    MathSciNet  Google Scholar 

  11. Srivastava, H.M., Fernandez, A., Baleanu, D.: Some new fractional-calculus connections between Mittag-Leffler functions. Mathematics 7(6), 485 (2019)

    Google Scholar 

  12. Iqbal, N., Niazi, A.U.K., Shafqat, R., Zaland, S.: Existence and uniqueness of mild solution for fractional-order controlled fuzzy evolution equation. J. Funct. Spaces 2021, 1–8 (2021)

    MathSciNet  Google Scholar 

  13. Devi, A., Kumar, A., Baleanu, D., Khan, A.: On stability analysis and existence of positive solutions for a general non-linear fractional differential equations. Adv. Differ. Equ. 2020(1), 1–16 (2020)

    MathSciNet  Google Scholar 

  14. Tajadodi, H., Khan, A., Francisco Gómez-Aguilar, J., Khan, H.: Optimal control problems with Atangana–Baleanu fractional derivative. Optim. Control Appl. Methods 42(1), 96–109 (2021)

    MathSciNet  Google Scholar 

  15. Giusti, A., Colombaro, I., Garra, R., Garrappa, R., Polito, F., Popolizio, M., Mainardi, F.: A practical guide to Prabhakar fractional calculus. Fract. Calc. Appl. Anal. 23(1), 9–54 (2020)

    MathSciNet  Google Scholar 

  16. Fernandez, A., Baleanu, D.: Classes of operators in fractional calculus: a case study. Math. Methods Appl. Sci. 44(11), 9143–9162 (2021)

    ADS  MathSciNet  Google Scholar 

  17. Raza, A., Thumma, T., Al-Khaled, K., Khan, S.U., Ghachem, K., Alhadri, M., Kolsi, L.: Prabhakar fractional model for viscous transient fluid with heat and mass transfer and Newtonian heating applications. Waves Random Compl. Media 33(3), 808–824 (2023)

    ADS  MathSciNet  Google Scholar 

  18. Eshaghi, S., Ghaziani, R.K., Ansari, A.: Stability and dynamics of neutral and integro-differential regularized Prabhakar fractional differential systems. Comput. Appl. Math. 39, 1–21 (2020)

    MathSciNet  Google Scholar 

  19. Garrappa, R., Mainardi, F., Guido, M.: Models of dielectric relaxation based on completely monotone functions. Fract. Calc. Appl. Anal. 19(5), 1105–1160 (2016)

    MathSciNet  Google Scholar 

  20. Giusti, A., Colombaro, I.: Prabhakar-like fractional viscoelasticity. Commun. Nonlinear Sci. Numer. Simul. 56, 138–143 (2018)

    ADS  MathSciNet  Google Scholar 

  21. Xu, J.: Time-fractional particle deposition in porous media. J. Phys. A: Math. Theor. 50(19), 195002 (2017)

    ADS  MathSciNet  Google Scholar 

  22. Zhang, J., Raza, A., Khan, U., Ali, Q., Zaib, A., Weera, W., Galal, A.M.: Thermophysical study of Oldroyd-b hybrid nanofluid with sinusoidal conditions and permeability: a Prabhakar fractional approach. Fractal Fract. 6(7), 357 (2022)

    Google Scholar 

  23. Hussain, S., Madi, E.N., Iqbal, N., Botmart, T., Karaca, Y., Mohammed, W.W.: Fractional dynamics of vector-borne infection with sexual transmission rate and vaccination. Mathematics 9(23), 3118 (2021)

    Google Scholar 

  24. Fazli, H., Sun, H., Nieto, J.J.: Fréchet-kolmogorov compactness of Prabhakar integral operator. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 115(4), 165 (2021)

    Google Scholar 

  25. Area, I., Nieto, J.J.: Fractional-order logistic differential equation with Mittag–Leffler-type kernel. Fractal Fract. 5(4), 273 (2021)

    Google Scholar 

  26. Garra, R., Garrappa, R.: The Prabhakar or three parameter Mittag-Leffler function: theory and application. Commun. Nonlinear Sci. Numer. Simul. 56, 314–329 (2018)

    ADS  MathSciNet  Google Scholar 

  27. Wang, Y., Mansir, I.B., Al-Khaled, K., Raza, A., Khan, S.U., Khan, M.I., Ahmed, A.E.-S.: Thermal outcomes for blood-based carbon nanotubes (swcnt and mwcnts) with Newtonian heating by using new prabhakar fractional derivative simulations. Case Stud. Thermal Eng. 32, 101904 (2022)

    Google Scholar 

  28. Balachandran, K., Park, J., Trujillo, J.: Controllability of nonlinear fractional dynamical systems. Nonlinear Anal. Theory Methods Appl. 75(4), 1919–1926 (2012)

    MathSciNet  Google Scholar 

  29. Sheng, J., Jiang, W., Pang, D., Wang, S.: Controllability of nonlinear fractional dynamical systems with a Mittag-Leffler kernel. Mathematics 8(12), 2139 (2020)

    Google Scholar 

  30. Balachandran, K., Govindaraj, V., Rodríguez-Germá, L., Trujillo, J.J.: Controllability results for nonlinear fractional-order dynamical systems. J. Optim. Theory Appl. 156, 33–44 (2013)

    MathSciNet  Google Scholar 

  31. Balachandran, K., Govindaraj, V., Rivero, M., Trujillo, J.J.: Controllability of fractional damped dynamical systems. Appl. Math. Comput. 257, 66–73 (2015)

    MathSciNet  Google Scholar 

  32. Muslim, M., George, R.K.: Trajectory controllability of the nonlinear systems governed by fractional differential equations. Differ. Equ. Dyn. Syst. 27, 529–537 (2019)

    MathSciNet  Google Scholar 

  33. Wang, J., Ahmed, H.M.: Null controllability of nonlocal Hilfer fractional stochastic differential equations. Miskolc Math. Notes 18(2), 1073–1083 (2017)

    MathSciNet  Google Scholar 

  34. Ahmed, H.M., Ragusa, M.A.: Nonlocal controllability of sobolev-type conformable fractional stochastic evolution inclusions with clarke subdifferential. Bull. Malays. Math. Sci. Soc. 45(6), 3239–3253 (2022)

    MathSciNet  Google Scholar 

  35. Bedi, P., Kumar, A., Abdeljawad, T., Khan, A.: Study of Hilfer fractional evolution equations by the properties of controllability and stability. Alex. Eng. J. 60(4), 3741–3749 (2021)

    Google Scholar 

  36. Bedi, P., Kumar, A., Abdeljawad, T., Khan, Z.A., Khan, A.: Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators. Adv. Differ. Equ. 2020(1), 1–15 (2020)

    MathSciNet  Google Scholar 

  37. Muslim, M., Kumar, A.: Controllability of fractional differential equation of order \(\alpha \in \)(1, 2] with non-instantaneous impulses. Asian J. Control 20(2), 935–942 (2018)

    MathSciNet  Google Scholar 

  38. Malik, M., Dhayal, R., Abbas, S.: Exact controllability of a retarded fractional differential equation with non-instantaneous impulses. Dyn. Cont. Dis. Ser. B 26, 53–69 (2019)

    MathSciNet  Google Scholar 

  39. Klamka, J.: Controllability of dynamical systems: a survey. Bull. Pol. Acad. Sci. Tech. Sci. 61(2) (2013)

  40. Kumar, V., Malik, M.: Controllability results of fractional integro-differential equation with non-instantaneous impulses on time scales. IMA J. Math. Control. Inf. 38(1), 211–231 (2021)

    MathSciNet  CAS  Google Scholar 

  41. Kavitha, K., Vijayakumar, V., Udhayakumar, R., Sakthivel, N., Sooppy Nisar, K.: A note on approximate controllability of the hilfer fractional neutral differential inclusions with infinite delay. Math. Methods Appl. Sci. 44(6), 4428–4447 (2021)

    ADS  MathSciNet  Google Scholar 

  42. Dhayal, R., Malik, M., Abbas, S.: Approximate and trajectory controllability of fractional stochastic differential equation with non-instantaneous impulses and poisson jumps. Asian J. Control 23(6), 2669–2680 (2021)

    MathSciNet  Google Scholar 

  43. Camacho, O., Leiva, H., Riera-Segura, L.: Controllability of semilinear neutral differential equations with impulses and nonlocal conditions. Math. Methods Appl. Sci. 45(16), 9826–9839 (2022)

    ADS  MathSciNet  Google Scholar 

  44. Zhou, Y., He, J.W.: New results on controllability of fractional evolution systems with order \(\alpha \in \)(1, 2). Evol. Equ. Control Theory 10(3), 491–509 (2021)

    MathSciNet  Google Scholar 

  45. Niazi, A.U.K., Iqbal, N., Mohammed, W.W.: Optimal control of nonlocal fractional evolution equations in the \(\alpha \)-norm of order (1,2). Adv. Differ. Equ. 2021(1), 142 (2021)

    MathSciNet  Google Scholar 

  46. Niazi, A.U.K., Iqbal, N., Shah, R., Wannalookkhee, F., Nonlaopon, K.: Controllability for fuzzy fractional evolution equations in credibility space. Fractal Fract. 5(3), 112 (2021)

    Google Scholar 

  47. He, J.W., Liang, Y., Ahmad, B., Zhou, Y.: Nonlocal fractional evolution inclusions of order \(\alpha \in \)(1, 2). Mathematics 7(2), 209 (2019)

    Google Scholar 

  48. Bedi, P., Kumar, A., Khan, A.: Controllability of neutral impulsive fractional differential equations with Atangana–Baleanu–Caputo derivatives. Chaos, Solitons Fractals 150, 111153 (2021)

    MathSciNet  Google Scholar 

  49. Srivastava, H.M., Choi, J.: Zeta and q-zeta functions and associated series and integrals (2011)

  50. Dauer, J.P.: Nonlinear perturbations of quasi-linear control systems. J. Math. Anal. Appl. 54(3), 717–725 (1976)

    MathSciNet  Google Scholar 

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Acknowledgements

Md. Samshad Hussain Ansari, the first author, would like to express his gratitude to the Council for Scientific and Industrial Research (CSIR), India, for the Ph.D. fellowship, file no. 09/1058(0026)/2021-EMR-I.

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Authors and Affiliations

  1. School of Mathematical & Statistical Sciences, Indian Institute of Technology Mandi, Kamand, Mandi, Himanchal Pradesh, 175005, India

    Md Samshad Hussain Ansari & Muslim Malik

  2. Institute of Space Sciences, P.O. Box MG-23, R 76900, Magurele-Bucharest, Romania

    Dumitru Baleanu

  3. Department of Medical Research, China Medical University, Taichung, 40447, Taiwan

    Dumitru Baleanu

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  1. Md Samshad Hussain Ansari
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MdSHA–Writing, Editing, Conceptualization, Methodology, MATLAB MM–Writing, Editing, Conceptualization, Methodology, MATLAB, Supervision DB-Writing, Editing, Conceptualization. All authors reviewed the manuscript.

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Correspondence to Muslim Malik.

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Ansari, M.S.H., Malik, M. & Baleanu, D. Controllability of Prabhakar Fractional Dynamical Systems. Qual. Theory Dyn. Syst. 23, 63 (2024). https://doi.org/10.1007/s12346-023-00919-4

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