Download PDF - Existence and controllability of fractional-order impulsive stochastic system with infinite delayAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Existence and controllability of fractional-order impulsive stochastic system with infinite delay

Authors 1

Affiliations

  1. Laboratory of Stochastic Models, Statistic and Applications, Tahar Moulay University, PO. Box 138 En-Nasr, 20000 Saida, Algeria

Abstract

This paper is concerned with the existence and approximate controllability for impulsive fractional-order stochastic infinite delay integro-differential equations in Hilbert space. By using Krasnoselskii's fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of impulsive fractional stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the obtained theory.

Keywords

ENG
existence result, approximate controllability, fractional stochastic differential equations, resolvent operators, infinite delay

Bibliography

  1. N. Abada, M. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Diff. Equ. 246 (2009), 3834-3863. doi: 10.1016/j.jde.2009.03.004
  2. P. Balasubramaniam, S.K. Ntouyas and D. Vinayagam, Existence of solutions of semilinear stochastic delay evolution inclusions in a Hilbert space, J. Math. Anal. Appl. 305 (2005) 438-451. doi: 10.1016/j.jmaa.2004.10.063
  3. J. Dabas, A. Chauhan and M. Kumar, Existence of the mild solutions for impulsive fractional equations with infinite delay, Int. J. Differ. Equ. 20 (2011). Article ID 793023.
  4. X. Fu and K. Mei, Approximate controllability of semilinear partial functional differential systems, J. Dynam. Control Syst. 15 (2009) 425-443. doi: 10.1007/s10883-009-9068-x
  5. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006.
  6. J. Klamka, Constrained controllability of nonlinear systems, J. Math. Anal. and Appl. 201 (2) (1996), 365-374. doi: 10.1006/jmaa.1996.0260
  7. J. Klamka, Constrained approximate boundary controllability, IEEE Transactions on Automatic Control AC-42 (2) (1997), 280-284. doi: 10.1109/9.554411
  8. J. Klamka, Schauder fixed point theorem in nonlinear controllability problems, Control Cybernet. 29 (2000), 153-165.
  9. J. Klamka, Constrained exact controllability of semilinear systems, Systems and Control Letters 4 (2) (2002), 139-147. doi: 10.1016/S0167-6911(02)00184-6
  10. J. Klamka, Stochastic controllability of systems with multiple delays in control, Int. J. Applied Math. Computer Sci. 19 (2009), 39-47. doi: 10.2478/v10006-009-0003-9
  11. J. Klamka, Constrained controllability of semilinear systems with delays, Nonlinear Dynamics 56 (2009), 169-177. doi: 10.1007/s11071-008-9389-4
  12. N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim. 42 (2003), 1604-1622. doi: 10.1137/S0363012901391688
  13. N.I. Mahmudov and A. Denker, On controllability of linear stochastic systems, Int. J. Control 73 (2000), 144-151. doi: 10.1080/002071700219849
  14. K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993.
  15. P. Muthukumar and P. Balasubramaniam, Approximate controllability of mixed stochastic Volterra-Fredholm type integrodifferential systems in Hilbert space, J. Franklin Inst. 348 (2011), 2911-2922. doi: 10.1016/j.jfranklin.2011.10.001
  16. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  17. Y. Ren, Q. Zhou and L. Chen, Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay, J. Optim. Theory Appl. 149 (2011), 315-331. doi: 10.1007/s10957-010-9792-0
  18. J. Sabatier, O.P. Agrawal and J.A. Tenreiro-Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer-Verlag, New York, 2007. doi: 10.1007/978-1-4020-6042-7
  19. R. Sakthivel and E. R. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Int. J. Control. 83 (2010), 387-393. doi: 10.1080/00207170903171348
  20. R. Sakthivel, Y. Ren and N.I. Mahmudov, Approximate controllability of second-order stochastic differential equation with impulsive effects, Modern Phys. Lett. (B) 24 (2010), 1559-1572. doi: 10.1142/S0217984910023359
  21. R. Sakthivel, J.J. Nieto and N.I. Mahmudov, Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay, Taiwanese J. Math. 14 (2010), 1777-1797.
  22. R. Sakthivel, S. Suganya and S.M. Anthoni, Approximate controllability of fractional stochastic evolution equations, Computers & Mathematics with Applications 63 (2012), 660-668. doi: 10.1016/j.camwa.2011.11.024
  23. R. Sakthivel, P. Revathi and Yong Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlin. Anal. (2012). doi: 10.1016/j.na.2012.10.009.
  24. L. Shen and J. Sun, Relative controllability of stochastic nonlinear systems with delay in control, Nonlin. Anal. RWA 13 (2012), 2880-2887. doi: 10.1016/j.nonrwa.2012.04.017
  25. X.B. Shu, Y. Lai and Y. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlin. Anal. TMA 74 (2011), 2003-2011. doi: 10.1016/j.na.2010.11.007
  26. L.W. Wang, Approximate controllability for integrodifferential equations with multiple delays, J. Optim. Theory Appl. 143 (2009), 185-206. doi: 10.1007/s10957-009-9545-0
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
2013/33/1
Export to PBN, Opens in new tab
Pages:
65-87
Main language of publication
English
Received
2013-02-25
Published
2013

DOI: 10.7151/dmdico1144

Exact and natural sciences