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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Reachability of Fractional Dynamical Systems with Single Delay in Control using $psi$-Hilfer Pseudo-Fractional Derivative

Journal of Applied Nonlinear Dynamics 13(3) (2024) 545--556 | DOI:10.5890/JAND.2024.09.010

A. Panneer Selvam, V. Govindaraj

Department of Mathematics, National Institute of Technology Puducherry, Karaikal - 609 609, India

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Abstract

In this article, we study the reachability of linear and non-linear fractional dynamical systems with single delay in control in the sense of $\psi$-Hilfer pseudo-fractional derivative. The necessary and sufficient conditions for reachability of linear fractional dynamical systems are obtained using Grammian matrix which is expressed by the Mittag-Leffler functions (one or two parameters). Sufficient conditions for reachability of nonlinear fractional dynamical systems are obtained by using Schauder's fixed point theorem. Two numerical examples are offered to help better understand of theoretical results.

Acknowledgments

A. Panneer Selvam would like to thank the University Grants Commission (UGC), New Delhi110002, for funding the research under the NET-SRF Scheme (Student ID: 191620040066) and V. Govindaraj would like to thank the National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Government of India, for funding the research project (File No. 02011/18/2023 NBHM (R.P)/ R\&D II/5952).

References

  1. [1]  Sousa, J.V.D.C. and De Oliveira, E.C. (2018), On the $\psi$-Hilfer fractional derivative, Communications in Nonlinear Science and Numerical Simulation, 60, 72-91.
  2. [2]  Sousa, J.V.D.C. and De Oliveira, E.C. (2019), Leibniz type rule: $\psi$-Hilfer fractional operator, Communications in Nonlinear Science and Numerical Simulation, 77, 305-311.
  3. [3]  Sousa, J.V.D.C., Frederico, G.S., and De Oliveira, E.C. (2020), $\psi$Hilfer pseudo-fractional operator: new results about fractional calculus, Computational and Applied Mathematics, 39(4), 254.
  4. [4]  Sousa, J.V.D.C., Vellappandi, M., Govindaraj, V., and Frederico, G.S. (2021), Reachability of fractional dynamical systems using $\psi$ Hilfer pseudo-fractional derivative, Journal of Mathematical Physics, 62(8), 082703.
  5. [5]  Wei, J. (2012), The controllability of fractional control systems with control delay, Computers $\&$ Mathematics with Applications, 64(10), 3153-3159.
  6. [6]  Zhang, H., Cao, J., and Jiang, W. (2013), Reachability and controllability of fractional singular dynamical systems with control delay, Journal of Applied Mathematics, 2013, 1-10.
  7. [7]  Muni, V.S., Govindaraj, V., and George, R.K. (2018), Controllability of fractional order semilinear systems with a delay in control, Indian Journal of Mathematics, 60(2), 311-335.
  8. [8]  Sikora, B. and Klamka, J. (2017), Constrained controllability of fractional linear systems with delays in control, Systems $\&$ Control Letters, 106, 9-15.
  9. [9]  Klamka, J. (2022), Controllability of fractional linear systems with delays in control, In Fractional Dynamical Systems: Methods, Algorithms and Applications, 307-330.
  10. [10]  Balachandran, K., Kokila, J., and Trujillo, J.J. (2012), Relative controllability of fractional dynamical systems with multiple delays in control, Computers $\&$ Mathematics with Applications, 64(10), 3037-3045.
  11. [11]  Trzasko, W. (2008), Reachability and controllability of positive fractional discrete-time systems with delay, Journal of Automation, Mobile Robotics $\&$ Intelligent Systems, 43-47.
  12. [12]  Panneer Selvam, A. and Govindaraj, V. (2022), Reachability of fractional dynamical systems with multiple delays in control using $\psi$-Hilfer pseudo-fractional derivative, Journal of Mathematical Physics, 63(10), 102706.
  13. [13]  Sajewski, L. (2016), Reachability, observability and minimum energy control of fractional positive continuous-time linear systems with two different fractional orders, Multidimensional Systems and Signal Processing, 27, 27-41.
  14. [14]  Rogowski, K. (2016), Reachability of standard and fractional continuous-time systems with constant inputs, Archives of Control Sciences, 26(2), 147-159.
  15. [15]  Sousa, J.V.D.C., De Oliveira, E.C., and Kucche, K.D. (2019), On the fractional functional differential equation with abstract Volterra operator, Bulletin of the Brazilian Mathematical Society, 50(4), 803-822.
  16. [16]  Panneer Selvam, A., Vellappandi, M., and Govindaraj, V. (2023), Controllability of fractional dynamical systems with $\psi$-Caputo fractional derivative, Physica Scripta, 98(2), 025206.
  17. [17]  Kaczorek, T. (2008), Fractional positive continuous-time linear systems and their reachability, International Journal of Applied Mathematics and Computer Science, 18(2), 223-228.
  18. [18]  Kaczorek, T. (2009), Reachability of cone fractional continuous-time linear systems, International Journal of Applied Mathematics and Computer Science, 19(1), 89-94.
  19. [19]  Poovarasan, R., Kumar, P., Nisar, K.S., and Govindaraj, V. (2023), The existence, uniqueness, and stability analyses of the generalized Caputo-type fractional boundary value problems, AIMS Mathematics, 8(7), 16757-16772.
  20. [20]  Kaczorek, T. (2016), Reachability Of fractional continuous-time linear systems using the Caputo-Fabrizio derivative, $30^{\rm th$ European Conference on Modelling and Simulation}, 53-58.
  21. [21]  Jose, S.A., Ramachandran, R., Baleanu, D., Panigoro, H.S., Alzabut, J., and Balas, V.E. (2023), Computational dynamics of a fractional order substance addictions transfer model with Atangana‐Baleanu‐Caputo derivative, Mathematical Methods in the Applied Sciences, 46(5), 5060-5085.
  22. [22]  Babakhani, A., Yadollahzadeh, M., and Neamaty, A. (2018), Some properties of pseudo-fractional operators, Journal of Pseudo-Differential Operators and Applications, 9, 677-700.
  23. [23]  Hosseini, M., Babakhani, A., Agahi, H., and Rasouli, S.H. (2016), On pseudo-fractional integral inequalities related to Hermite–Hadamard type, Soft Computing, 20, 2521-2529.
  24. [24]  Pap, E. (2008), Pseudo-additive measures and their applications, Handbook of Measure Theory, 1403-1468.
  25. [25]  Pap, E. (2005), Applications of the generated pseudo-analysis to nonlinear partial differential equations, Contemporary Mathematics, 377, 239-260.
  26. [26]  Dauer, J.P. (1976), Nonlinear perturbations of quasi-linear control problems, Journal of Mathematical Analysis and Applications, 54, 717-725.

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