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On the controllability of nonlinear fractional system with control delay

Year 2020, Volume: 49 Issue: 1, 294 - 302, 06.02.2020
https://doi.org/10.15672/hujms.546990

Abstract

We discuss the controllability of nonlinear fractional control system with control delay. Firstly we obtain result about controllability of a linear fractional control system. After that, we give sufficient condition for the controllability of nonlinear fractional system with control delay. Our approach is based on Schauder fixed point theorem. At the end numerical example is constructed to support the result.

References

  • [1] J.O. Alzabut, Existence of periodic solutions of a type of nonlinear impulsive delay differential equations with a small parameter, J. Nonlinear Math. Phys. 15, 13–21, 2008.
  • [2] K. Balachandran, J.P. Dauer, Controllability of nonlinear systems via fixed-point theorems, J. Optim. Theory Appl. 53 (3), 345–352, 1987.
  • [3] K. Balachandran, V. Govindaraj, L. Rodriguez-Germa, and J.J. Trujillo, Controllability of nonlinear higher order fractional dynamical systems, Nonlinear Dynam. 156, 33–44, 2013.
  • [4] K. Balachandran, J.Y. Park, and J.J. Trujillo, Controllability of nonlinear fractional dynamical systems, Nonlinear Anal. 75 (4), 1919–1926, 2012.
  • [5] L. Dai, Singular Control Systems, Springer, 1989.
  • [6] J.P. Dauer, Nonlinear Perturbations of Quasi-Linear Control Systems, J. Math. Anal. Appl. 54, 717–725, 1976.
  • [7] J.P. Dauer, R.D. Gahl, Controllability of nonlinear delay systems, J. Optimiz. Theory Appl. 21 (1), 59–70, 1977.
  • [8] Park, K. Diethel, The analysis of fractional differential equations, Lect. Notes Math., 2010.
  • [9] J. Hale, Introduction to functional differential equations, Springer Verlag, 1992.
  • [10] W. Jiang, The degeneration differential systems with delay, Anhui University Press, 1998.
  • [11] W. Jiang, Eigenvalue and stability of singular differential delay systems, J. Math. Anal. Appl. 297, 305–316, 2004.
  • [12] W. Jiang, Function-controllability of nonlinear singular delay differential control systems, Acta Math. Sinica (Chin. Ser.) 49 (5), 1153–1162, 2006.
  • [13] W. Jiang, On the solvability of singular differential delay systems with variable coefficients, Int. J. Dyn. Syst. Differ. Equ. 4, 245–249, 2008.
  • [14] W. Jiang, The constant variation formulae for singular fractional differential systems with delay, Comput. Math. Appl. 59 (3), 1184–1190, 2010.
  • [15] W. Jiang, The controllability of fractional control systems with control delay, Comput. Math. Appl. 64, 3153–3159, 2012.
  • [16] W. Jiang, On the interval controllability of fractional systems with control delay, J. Math. Res. 9 (5), 87, 2017.
  • [17] W. Jiang and W. Song, Controllability of singular systems with control delay, Automatica J. IFAC 37, 1873–1877, 2001.
  • [18] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Publishers, 204, 2006.
  • [19] K.S. Miller and B. Ross, An introduction to the fractional calculus and differential equations, John Wiley and Sons, 1993.
  • [20] R.J. Nirmala, K. Balachandran, L.R. Germa, and J.J. Trujillo, Controllability of nonlinear fractional delay dynamical systems, Rep. Math. Phys. 77 (1), 87–104, 2016.
  • [21] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
Year 2020, Volume: 49 Issue: 1, 294 - 302, 06.02.2020
https://doi.org/10.15672/hujms.546990

Abstract

References

  • [1] J.O. Alzabut, Existence of periodic solutions of a type of nonlinear impulsive delay differential equations with a small parameter, J. Nonlinear Math. Phys. 15, 13–21, 2008.
  • [2] K. Balachandran, J.P. Dauer, Controllability of nonlinear systems via fixed-point theorems, J. Optim. Theory Appl. 53 (3), 345–352, 1987.
  • [3] K. Balachandran, V. Govindaraj, L. Rodriguez-Germa, and J.J. Trujillo, Controllability of nonlinear higher order fractional dynamical systems, Nonlinear Dynam. 156, 33–44, 2013.
  • [4] K. Balachandran, J.Y. Park, and J.J. Trujillo, Controllability of nonlinear fractional dynamical systems, Nonlinear Anal. 75 (4), 1919–1926, 2012.
  • [5] L. Dai, Singular Control Systems, Springer, 1989.
  • [6] J.P. Dauer, Nonlinear Perturbations of Quasi-Linear Control Systems, J. Math. Anal. Appl. 54, 717–725, 1976.
  • [7] J.P. Dauer, R.D. Gahl, Controllability of nonlinear delay systems, J. Optimiz. Theory Appl. 21 (1), 59–70, 1977.
  • [8] Park, K. Diethel, The analysis of fractional differential equations, Lect. Notes Math., 2010.
  • [9] J. Hale, Introduction to functional differential equations, Springer Verlag, 1992.
  • [10] W. Jiang, The degeneration differential systems with delay, Anhui University Press, 1998.
  • [11] W. Jiang, Eigenvalue and stability of singular differential delay systems, J. Math. Anal. Appl. 297, 305–316, 2004.
  • [12] W. Jiang, Function-controllability of nonlinear singular delay differential control systems, Acta Math. Sinica (Chin. Ser.) 49 (5), 1153–1162, 2006.
  • [13] W. Jiang, On the solvability of singular differential delay systems with variable coefficients, Int. J. Dyn. Syst. Differ. Equ. 4, 245–249, 2008.
  • [14] W. Jiang, The constant variation formulae for singular fractional differential systems with delay, Comput. Math. Appl. 59 (3), 1184–1190, 2010.
  • [15] W. Jiang, The controllability of fractional control systems with control delay, Comput. Math. Appl. 64, 3153–3159, 2012.
  • [16] W. Jiang, On the interval controllability of fractional systems with control delay, J. Math. Res. 9 (5), 87, 2017.
  • [17] W. Jiang and W. Song, Controllability of singular systems with control delay, Automatica J. IFAC 37, 1873–1877, 2001.
  • [18] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Publishers, 204, 2006.
  • [19] K.S. Miller and B. Ross, An introduction to the fractional calculus and differential equations, John Wiley and Sons, 1993.
  • [20] R.J. Nirmala, K. Balachandran, L.R. Germa, and J.J. Trujillo, Controllability of nonlinear fractional delay dynamical systems, Rep. Math. Phys. 77 (1), 87–104, 2016.
  • [21] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Musarrat Nawaz 0000-0001-6418-1283

Jiang Wei This is me 0000-0002-5446-1860

Jiale Sheng This is me 0000-0003-0207-9712

Azmat Ullah Khan Niazi This is me 0000-0001-9114-8075

Lichang Yang This is me 0000-0002-5555-3433

Publication Date February 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 1

Cite

APA Nawaz, M., Wei, J., Sheng, J., Ullah Khan Niazi, A., et al. (2020). On the controllability of nonlinear fractional system with control delay. Hacettepe Journal of Mathematics and Statistics, 49(1), 294-302. https://doi.org/10.15672/hujms.546990
AMA Nawaz M, Wei J, Sheng J, Ullah Khan Niazi A, Yang L. On the controllability of nonlinear fractional system with control delay. Hacettepe Journal of Mathematics and Statistics. February 2020;49(1):294-302. doi:10.15672/hujms.546990
Chicago Nawaz, Musarrat, Jiang Wei, Jiale Sheng, Azmat Ullah Khan Niazi, and Lichang Yang. “On the Controllability of Nonlinear Fractional System With Control Delay”. Hacettepe Journal of Mathematics and Statistics 49, no. 1 (February 2020): 294-302. https://doi.org/10.15672/hujms.546990.
EndNote Nawaz M, Wei J, Sheng J, Ullah Khan Niazi A, Yang L (February 1, 2020) On the controllability of nonlinear fractional system with control delay. Hacettepe Journal of Mathematics and Statistics 49 1 294–302.
IEEE M. Nawaz, J. Wei, J. Sheng, A. Ullah Khan Niazi, and L. Yang, “On the controllability of nonlinear fractional system with control delay”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, pp. 294–302, 2020, doi: 10.15672/hujms.546990.
ISNAD Nawaz, Musarrat et al. “On the Controllability of Nonlinear Fractional System With Control Delay”. Hacettepe Journal of Mathematics and Statistics 49/1 (February 2020), 294-302. https://doi.org/10.15672/hujms.546990.
JAMA Nawaz M, Wei J, Sheng J, Ullah Khan Niazi A, Yang L. On the controllability of nonlinear fractional system with control delay. Hacettepe Journal of Mathematics and Statistics. 2020;49:294–302.
MLA Nawaz, Musarrat et al. “On the Controllability of Nonlinear Fractional System With Control Delay”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, 2020, pp. 294-02, doi:10.15672/hujms.546990.
Vancouver Nawaz M, Wei J, Sheng J, Ullah Khan Niazi A, Yang L. On the controllability of nonlinear fractional system with control delay. Hacettepe Journal of Mathematics and Statistics. 2020;49(1):294-302.