[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
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

Existence and Controllability Results for Higher Order Caputo Fractional Damping System with Impulses

Journal of Applied Nonlinear Dynamics 12(4) (2023) 609--630 | DOI:10.5890/JAND.2023.12.001

M. Tamilarasi$^1$, B. Radhakrishnan$^2$

$^1$ Department of Mathematics, Sri Eshwar College of Engineering, Coimbatore - 641202, Tamil Nadu, India

$^2$ Department of Mathematics, PSG College of Technology, Coimbatore 641004, Tamil Nadu, India

Download Full Text PDF

 

Abstract

This paper investigates the presence of mild solutions and controllability for higher order Caputo fractional damped differential systems with impulses. The general method of Laplace transform, Grammian matrix, and sequential approximation techniques are used to establish the controllability requirement for the linear and nonlinear fractional damped dynamical systems of higher order. At the end, an example is provided to validate the theoretical results.

References

  1. [1]  Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley: New York.
  2. [2]  Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., and Feliu-Batlle, V. (2010), Fractional - order Systems and Controls, Fundamentals and Applications, Springer Science \& Business Media: New York.
  3. [3]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
  4. [4]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press: New York.
  5. [5]  Podlubny, I. (2002), Geometric \& physical interpretation of fractional integration fractional differentiation, Fractional Calculus Applications $\&$ Analysis, 5, 367-386.
  6. [6]  Kalman, R.E., Ho, Y.C., and Narendra, K.S. (1963), Controllability of linear dynamical system, Contributions to Differential Equations, 2, 189-213.
  7. [7]  Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag: New York.
  8. [8]  Radhakrishnan, B. and Balachandran, K. (2011), Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay, Nonlinear Analysis: Hybrid Systems, 5, 655-670.
  9. [9]  Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S, (1989), Theory of Impulsive Differential Equations, World Scientific: Singapore.
  10. [10]  Samoilenko, A.M. and Perestyuk, N.A. (1995), Impulsive Differential Equations, World Scientific: Singapore.
  11. [11]  Narahari Achar, B.N., Hanneken, J.W., Enck, T., and Clarke, T. (2001), Dynamics of the fractional oscillator, Physica A, 297(3–4), 361-367.
  12. [12]  Narahari Achar, B.N., Hanneken, J.W., and Clarke, T. (2002), Response characteristics of a fractional oscillator, Physica A, 309(3–4), 275-288.
  13. [13]  Tofighi, A. (2003), The intrinsic damping of the fractional oscillator, Physica A, 329, 29-34.
  14. [14]  Stanislavsky, A.A. (2004), Fractional oscillations, Physical Review E, 70, 1-6.
  15. [15]  Stanislavsky, A.A. (2005), Twist of fractional oscillations, Physica A, 354, 101-110.
  16. [16]  Balachandran, K., Govindaraj, V., Rivero, M., and Trujillo, J.J. (2015), Controllability of fractional damped dynamical systems, Applied Mathematics and Computation, 257, 66–73.
  17. [17]  Radhakrishnan, B. and Tamilarasi, M. (2018), Existence of solutions for quasilinear random impulsive neutral differential evolution equation, Arab Journal of Mathematical Sciences, 24, 235-246.
  18. [18]  Radhakrishnan, B. and Tamilarasi, M. (2019), Existence results for quasilinear random impulsive abstract differential inclusions in Hilbert space, The Journal of Analysis, 27(2), 327-345.
  19. [19]  Radhakrishnan, B., Tamilarasi, M., and Anukokila, P. (2018), Existence, uniqueness and stability results for semilinear integrodifferential non-local evolution equations with random impulse, Filomat, 32(19), 6615-6626.
  20. [20]  Radhakrishnan, B. and Tamilarasi, M. (2021), Existence, Uniqueness and Stability Results for Fractional Hybrid Pantograph Equation with Random Impulse, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications $\&$ Algorithms, 28(3), 165-181.
  21. [21]  Radhakrishnan, B. and Tamilarasi, M. (2021), A new approach for the generalized fractional Casson fluid model with Newtonian heating described by the modified Riemann–Liouville fractional operator, Mathematical methods in the Applied sciences, https://doi.org/10.1002/mma.8004.
  22. [22]  Tamilarasi, M., Radhakrishnan, B., and Anukokila, P. (2022), Approximate Controllability of Fractional Semilinear Delay Differential Control System with Random Impulse, Palestine Journal of Mathematics(Special Issue I), 11, 141–150.
  23. [23]  Kavitha, K., Vijayakumar, V., Udhayakumar, R., Sakthivel, N., and Nisar, K.S. (2020), A note on approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay, Mathematical methods in the Applied sciences, https://doi.org/10.1002/mma.7040.
  24. [24]  MohanRaja, M., Vijayakumar, V., and Udhayakumar, R. (2020), A new approach on approximate controllability of fractional evolution inclusions of order $1 < r < 2$ with infinite delay, Chaos, Solitons $\&$ Fractals, 141, 110343.
  25. [25]  Zhou, Y., Vijayakumar, V., Ravichandran, C., and Murugesu, R. (2017), Controllability results for fractional order neutral functional differential inclusions with infinite delay, Fixed Point Theory, 18(2), 773-798.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates