On the controllability of nonlinear fractional system with control delay

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.

Keywords

• Controllability,nonlinear system,control delay

References

1. [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. [2] K. Balachandran, J.P. Dauer, Controllability of nonlinear systems via fixed-point theorems, J. Optim. Theory Appl. 53 (3), 345–352, 1987.
3. [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. [4] K. Balachandran, J.Y. Park, and J.J. Trujillo, Controllability of nonlinear fractional dynamical systems, Nonlinear Anal. 75 (4), 1919–1926, 2012.
5. [5] L. Dai, Singular Control Systems, Springer, 1989.
6. [6] J.P. Dauer, Nonlinear Perturbations of Quasi-Linear Control Systems, J. Math. Anal. Appl. 54, 717–725, 1976.
7. [7] J.P. Dauer, R.D. Gahl, Controllability of nonlinear delay systems, J. Optimiz. Theory Appl. 21 (1), 59–70, 1977.
8. [8] Park, K. Diethel, The analysis of fractional differential equations, Lect. Notes Math., 2010.

9. [9] J. Hale, Introduction to functional differential equations, Springer Verlag, 1992.
10. [10] W. Jiang, The degeneration differential systems with delay, Anhui University Press, 1998.
11. [11] W. Jiang, Eigenvalue and stability of singular differential delay systems, J. Math. Anal. Appl. 297, 305–316, 2004.
12. [12] W. Jiang, Function-controllability of nonlinear singular delay differential control systems, Acta Math. Sinica (Chin. Ser.) 49 (5), 1153–1162, 2006.
13. [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. [14] W. Jiang, The constant variation formulae for singular fractional differential systems with delay, Comput. Math. Appl. 59 (3), 1184–1190, 2010.
15. [15] W. Jiang, The controllability of fractional control systems with control delay, Comput. Math. Appl. 64, 3153–3159, 2012.
16. [16] W. Jiang, On the interval controllability of fractional systems with control delay, J. Math. Res. 9 (5), 87, 2017.
17. [17] W. Jiang and W. Song, Controllability of singular systems with control delay, Automatica J. IFAC 37, 1873–1877, 2001.
18. [18] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Publishers, 204, 2006.
19. [19] K.S. Miller and B. Ross, An introduction to the fractional calculus and differential equations, John Wiley and Sons, 1993.
20. [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. [21] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.