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

Boundary Controllability of Delay Differential Systems of Fractional Order with Nonlocal Condition

Journal of Applied Nonlinear Dynamics 6(4) (2017) 465--472 | DOI:10.5890/JAND.2017.12.002

Kamalendra Kumar$^{1}$; Rakesh Kumar$^{2}$

$^{1}$ Department of Mathematics, SRMS College of Engineering and Technology, Bareilly-243001, India

$^{2}$ Department of Mathematics, Hindu College, Moradabad-244001, India

Abstract

Sufficient conditions for boundary controllability of time varying delay differential systems of fractional order with nonlocal condition in Banach space are established. The results are obtained by using fixed point theorems. An example is provided to illustrate our results.

Acknowledgments

The authors wish to thank the referees for their valuable comments and suggestions.

References

1.  [1] Balachandran, K. and Chandrasekran, M. (1996), Existence of solutions of delay differential equation with nonlocal condition, Indian J. pure appl. Math., 27(5), 443-449.
2.  [2] Kumar, K. and Kumar, R. (2015), Controllability results for general integrodifferential evolution equations in Banach space, Differential Equation and Control Process, 2015(3), 1-15.
3.  [3] Abd El-Ghaffar, A., Moubarak, M.R.A., and Shamardan, A.B. (2000), Controllability of fractional nonlinear control system, Journal of Fractional Calculus, 17, 59-69.
4.  [4] Chen, Y.Q., Ahu, H.S., and Xue, D. (2006), Robust controllability of interval fractional order linear time invariant systems, Signal Processing, 86, 2794-2802.
5.  [5] Shamardan, A.B. and Moubarak, M.R.A. (1999), Controllability and observability for fractional control systems, Journal of Fractional Calculus, 15, 25-34.
6.  [6] Balachandran, K. and Divya, S. (2016), Relative Controllability of Nonlinear Neutral Fractional Volterra Integrodifferential Systems with Multiple Delays in Control, Journal of Applied Nonlinear Dynamics, 5(2), 147-160.
7.  [7] Balakrishnan, A.V. (1976), Applied Functional Analysis, Springer Verlag, New York.
8.  [8] Curtain, R.F. and Zwart, H.J. (1995), An Introduction to Infinite Dimensional Linear Systems Theory, Springer Verlag, New York.
9.  [9] Washburn, D. (1979), A bound on the boundary input map for parabolic equations with application to time optimal control, SIAM Journal on Control and Optimization, 17, 652-671.
10.  [10] Barbu, V. (1980), Boundary control problems with convex cost criterion, SIAM Journal of Control and optimization, 18, 227-243.
11.  [11] Fattorini, H.O. (1968) Boundary control systems, SIAM Journal on Control, 6, 349-384.
12.  [12] Lasiecka, I. (1978), Boundary control of parabolic systems; regularity of solutions, Applied Mathematics and optimization, 4, 301-327.
13.  [13] Balachandran, K. and Anandhi, E.R. (2001), Boundary controllability of integrodifferential systems in Banach spaces, Proc. Indian Acad. Sci. (Math. Sci.), 111(1), 127-135.
14.  [14] Balachandran, K. and Anandhi, E.R. (2000), Boundary controllability of delay integrodifferential systems in Banach spaces, J. KSIM, 4(2), 67-75.
15.  [15] Balachandran, K., Anandhi, E.R., and Dauer, J.P. (2003), Boundary controllability of Sobolev-type abstract nonlinear integrodifferential systems, J. Math. Anal. Appl., 277(2), 446-464.
16.  [16] Han, H.K. and Park, J.Y. (1999), Boundary controllability of differential equations with nonlocal condition, J. Math. Anal. Appl. 230(1), 242-250.
17.  [17] Park, J.Y. and Jeong, J.U. (2011), Boundary controllability of semilinear neutral evolution systems, Bull. Korean Math. Soc., 48(4), 705-712.
18.  [18] Ahmed, H.M. (2010), Boundary controllability of nonlinear fractional integrodifferential systems, Advances in Difference Equations, Volume 2010, Article ID 279493, 9 pages.
19.  [19] El-Borai,M.M. (2002), Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons and Fractals, 14(3), 433-440.
20.  [20] Gorenflo, R. and Mainardi, F. (1998), Fractional calculus and stable probability distributions, Archives of Mechanics, 50(3), 377-388.
21.  [21] El-Borai, M.M., El-Nadi, K.E., Mostafa, O.L., and Ahmed, H.M. (2006), Semigroup and some fractional stochastic integral equations, The International Journal of Pure and Applied Mathematical Sciences, 3(1), 47-52.
22.  [22] El-Borai, M.M., El-Nadi, K.E., Mostafa, O.L., and Ahmed, H.M. (2004), Volterra equations with fractional stochastic integrals, Mathematical problem in engineering, 2004(5), 453-468.
23.  [23] El-Borai,M.M. (2004), The fundamental solutions for fractional evolution equations of parabolic type, Journal of Applied Mathematics and Stochastic Analysis, 2004(3), 197-211.