Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 146010, 9 pages
http://dx.doi.org/10.1155/2013/146010
Research Article

Controllability Criteria for Linear Fractional Differential Systems with State Delay and Impulses

1Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China
2Department of Mathematics, Anqing Normal University, Anqing, Anhui 246133, China
3Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
4School of Mathematical Sciences, Anhui University, Hefei, Anhui 230039, China

Received 8 April 2013; Accepted 14 May 2013

Academic Editor: Francisco J. Marcellán

Copyright © 2013 Hai Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at Zentralblatt MATH · View at MathSciNet
  2. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Technical University of Kosice, Kosice, Slovak Republic, 1999. View at Zentralblatt MATH · View at MathSciNet
  3. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Z. M. Odibat, “Analytic study on linear systems of fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1171–1183, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 11, no. 5, pp. 4465–4475, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Wang and H. Xiang, “Upper and lower solutions method for a class of singular fractional boundary value problems with p-Laplacian operator,” Abstract and Applied Analysis, vol. 2010, Article ID 971824, 12 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. Deng and L. Ma, “Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 6, pp. 676–680, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. R. W. Ibrahim, “Complex transforms for systems of fractional differential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 814759, 15 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. K. Sayevand, A. Golbabai, and A. Yildirim, “Analysis of differential equations of fractional order,” Applied Mathematical Modelling, vol. 36, no. 9, pp. 4356–4364, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. W. Q. Wu and X. B. Zhou, “Eigenvalue of fractional differential equations with Laplacian operator,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 137890, 8 pages, 2013. View at Publisher · View at Google Scholar
  12. X. Zhang, “Some results of linear fractional order time-delay system,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 407–411, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for p-type fractional neutral differential equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 71, no. 7-8, pp. 2724–2733, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  14. F. Chen and Y. Zhou, “Attractivity of fractional functional differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1359–1369, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. E. Kaslik and S. Sivasundaram, “Analytical and numerical methods for the stability analysis of linear fractional delay differential equations,” Journal of Computational and Applied Mathematics, vol. 236, no. 16, pp. 4027–4041, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. D. Baĭnov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  17. M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2, Hindawi Publishing Corporation, New York, NY, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  18. M. Fečkan, Y. Zhou, and J. Wang, “On the concept and existence of solution for impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 3050–3060, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. R. E. Kalman, Y. C. Ho, and K. S. Narendra, “Controllability of linear dynamical systems,” Contributions to Differential Equations, vol. 1, no. 2, pp. 189–213, 1963. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. W. M. Wonham and A. S. Morse, “Decoupling and pole assignment in linear multivariable systems: a geometric approach,” SIAM Journal on Control and Optimization, vol. 8, pp. 1–18, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. R. Manzanilla, L. G. Mármol, and C. J. Vanegas, “On the controllability of a differential equation with delayed and advanced arguments,” Abstract and Applied Analysis, vol. 2010, Article ID 307409, 16 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. L. Wang, “Approximate boundary controllability for semilinear delay differential equations,” Journal of Applied Mathematics, vol. 2011, Article ID 587890, 10 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. H. Shi, G. Xie, and W. Luo, “Controllability analysis of linear discrete time systems with time delay in state,” Abstract and Applied Analysis, vol. 2012, Article ID 10.1155/2012/490903, 11 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. H. Shi, G. M. Xie, and W. G. Luo, “Controllability of linear discrete-time systems with both delayed states and delayed inputs,” Abstract and Applied Analysis, vol. 2013, Article ID 975461, 5 pages, 2013. View at Publisher · View at Google Scholar
  25. S. H. Chen, W. H. Ho, and J. H. Chou, “Robust local regularity and controllability of uncertain TS fuzzy descriptor systems,” Journal of Applied Mathematics, vol. 2012, Article ID 825416, 14 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  26. W. Jiang and W. Z. Song, “Controllability of singular systems with control delay,” Automatica, vol. 37, no. 11, pp. 1873–1877, 2001. View at Publisher · View at Google Scholar
  27. L. Zhang, Y. Ding, T. Wang, L. Hu, and K. Hao, “Controllability of second-order semilinear impulsive stochastic neutral functional evolution equations,” Mathematical Problems in Engineering, vol. 2012, Article ID 748091, 13 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  28. H. Shi and G. Xie, “Controllability and observability criteria for linear piecewise constant impulsive systems,” Journal of Applied Mathematics, vol. 2012, Article ID 182040, 24 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. X. J. Wan, Y. P. Zhang, and J. T. Sun, “Controllability of impulsive neutral functional differential inclusions in Banach spaces,” Journal of Applied Mathematics, vol. 2013, Article ID 861568, 8 pages, 2013. View at Publisher · View at Google Scholar
  30. S. A. Ammour, S. Djennoune, and M. Bettayeb, “A sliding mode control for linear fractional systems with input and state delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2310–2318, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. K. Balachandran, J. Y. Park, and J. J. Trujillo, “Controllability of nonlinear fractional dynamical systems,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 75, no. 4, pp. 1919–1926, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  32. K. Balachandran, J. Kokila, and J. J. Trujillo, “Relative controllability of fractional dynamical systems with multiple delays in control,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3037–3045, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  33. A. Debbouche and D. Baleanu, “Exact null controllability for fractional nonlocal integrodifferential equations via implicit evolution system,” Journal of Applied Mathematics, vol. 2012, Article ID 931975, 17 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. N. I. Mahmudov, “Approximate controllability of fractional Sobolev-type evolution equations in Banach spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 502839, 9 pages, 2013. View at Publisher · View at Google Scholar
  35. T. L. Guo, “Controllability and observability of impulsive fractional linear time-invariant system,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3171–3182, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  36. X. F. Zhou, W. Jiang, and L. G. Hu, “Controllability of a fractional linear time-invariant neutral dynamical system,” Applied Mathematics Letters, vol. 26, no. 4, pp. 418–424, 2013. View at Google Scholar