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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 262191, 10 pages
http://dx.doi.org/10.1155/2013/262191
Research Article

Approximate Controllability of Sobolev Type Nonlocal Fractional Stochastic Dynamic Systems in Hilbert Spaces

1Department of Mathematics, Guelma University, 24000 Guelma, Algeria
2Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
3Department of Mathematics and Computer Sciences, Cankaya University, 06530 Ankara, Turkey
4Institute of Space Sciences, Magurele, Bucharest, Romania

Received 19 July 2013; Accepted 27 September 2013

Academic Editor: Bashir Ahmad

Copyright © 2013 Mourad Kerboua 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. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3 of Complexity, Nonlinearity and Chaos, World Scientific, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  2. M. Bragdi, A. Debbouche, and D. Baleanu, “Existence of solutions for fractional differential inclusions with separated boundary conditions in Banach space,” Advances in Mathematical Physics, vol. 2013, Article ID 426061, 5 pages, 2013. View at Zentralblatt MATH · View at MathSciNet
  3. A. Debbouche, “Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems,” Advances in Difference Equations, vol. 2011, article 5, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Debbouche, D. Baleanu, and R. P. Agarwal, “Nonlocal Nonlinear Integro-Differential Equations of Fractional Orders,” Boundary Value Problems, no. 2012, article 78, 2012.
  5. A. M. A. El-Sayed, “Fractional-order diffusion-wave equation,” International Journal of Theoretical Physics, vol. 35, no. 2, pp. 311–322, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378, pp. 291–348, Springer, New York, NY, USA, 1997. View at MathSciNet
  7. A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, UK, 2012. View at MathSciNet
  8. N. Nyamoradi, D. Baleanu, and T. Bashiri, “Positive solutions to fractional boundary value problems with nonlinear boundary conditions,” Abstract and Applied Analysis, vol. 2013, Article ID 579740, 20 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  9. B. Ahmad and S. K. Ntouyas, “A note on fractional differential equations with fractional separated boundary conditions,” Abstract and Applied Analysis, vol. 2012, Article ID 818703, 11 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. Sakthivel, R. Ganesh, and S. Suganya, “Approximate controllability of fractional neutral stochastic system with infinite delay,” Reports on Mathematical Physics, vol. 70, no. 3, pp. 291–311, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Sakthivel, R. Ganesh, Y. Ren, and S. M. Anthoni, “Approximate controllability of nonlinear fractional dynamical systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 12, pp. 3498–3508, 2013.
  12. A. E. Bashirov and N. I. Mahmudov, “On concepts of controllability for deterministic and stochastic systems,” SIAM Journal on Control and Optimization, vol. 37, no. 6, pp. 1808–1821, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1442–1450, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. 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
  15. R. Sakthivel, N. I. Mahmudov, and J. J. Nieto, “Controllability for a class of fractional-order neutral evolution control systems,” Applied Mathematics and Computation, vol. 218, no. 20, pp. 10334–10340, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. Triggiani, “A note on the lack of exact controllability for mild solutions in Banach spaces,” SIAM Journal on Control and Optimization, vol. 15, no. 3, pp. 407–411, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. P. Dauer and N. I. Mahmudov, “Approximate controllability of semilinear functional equations in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 273, no. 2, pp. 310–327, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. N. Sukavanam and N. K. Tomar, “Approximate controllability of semilinear delay control systems,” Nonlinear Functional Analysis and Applications, vol. 12, no. 1, pp. 53–59, 2007. View at Zentralblatt MATH · View at MathSciNet
  19. W. Bian, “Approximate controllability for semilinear systems,” Acta Mathematica Hungarica, vol. 81, no. 1-2, pp. 41–57, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. E. N. Chukwu and S. M. Lenhart, “Controllability questions for nonlinear systems in abstract spaces,” Journal of Optimization Theory and Applications, vol. 68, no. 3, pp. 437–462, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. J.-M. Jeong, J.-R. Kim, and H.-H. Roh, “Controllability for semilinear retarded control systems in Hilbert spaces,” Journal of Dynamical and Control Systems, vol. 13, no. 4, pp. 577–591, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J.-M. Jeong and H.-G. Kim, “Controllability for semilinear functional integrodifferential equations,” Bulletin of the Korean Mathematical Society, vol. 46, no. 3, pp. 463–475, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. N. Sukavanam and Divya, “Approximate controllability of abstract semilinear deterministic control system,” Bulletin of the Calcutta Mathematical Society, vol. 96, no. 3, pp. 195–202, 2004. View at Zentralblatt MATH · View at MathSciNet
  24. S. Kumar and N. Sukavanam, “Approximate controllability of fractional order semilinear systems with bounded delay,” Journal of Differential Equations, vol. 252, no. 11, pp. 6163–6174, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. S. Rathinasamy and R. Yong, “Approximate controllability of fractional differential equations with state-dependent delay,” Results in Mathematics, vol. 63, no. 3-4, pp. 949–963, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. Sakthivel, Y. Ren, and N. I. Mahmudov, “On the approximate controllability of semilinear fractional differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1451–1459, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. N. Sukavanam and S. Kumar, “Approximate controllability of fractional order semilinear delay systems,” Journal of Optimization Theory and Applications, vol. 151, no. 2, pp. 373–384, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. J. Cao, Q. Yang, Z. Huang, and Q. Liu, “Asymptotically almost periodic solutions of stochastic functional differential equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1499–1511, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. Cao, Q. Yang, and Z. Huang, “On almost periodic mild solutions for stochastic functional differential equations,” Nonlinear Analysis. Real World Applications, vol. 13, no. 1, pp. 275–286, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Y.-K. Chang, Z.-H. Zhao, G. M. N'Guérékata, and R. Ma, “Stepanov-like almost automorphy for stochastic processes and applications to stochastic differential equations,” Nonlinear Analysis. Real World Applications, vol. 12, no. 2, pp. 1130–1139, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. X. Mao, Stochastic Differential Equations and Applications, Ellis Horwood, Chichester, UK, 1997.
  32. R. Sakthivel, P. Revathi, and Y. Ren, “Existence of solutions for nonlinear fractional stochastic differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 81, pp. 70–86, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. N. I. Mahmudov, “Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces,” SIAM Journal on Control and Optimization, vol. 42, no. 5, pp. 1604–1622, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. F. Li, J. Liang, and H. K. Xu, “Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions,” Journal of Mathematical Analysis and Applications., vol. 391, pp. 510–525, 2012.
  35. R. Sakthivel, S. Suganya, and S. M. Anthoni, “Approximate controllability of fractional stochastic evolution equations,” Computers & Mathematics with Applications, vol. 63, no. 3, pp. 660–668, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. M. Fečkan, J. Wang, and Y. Zhou, “Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators,” Journal of Optimization Theory and Applications, vol. 156, no. 1, pp. 79–95, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  38. 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, Amsterdam, The Netherlands, 2006. View at MathSciNet
  39. 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 MathSciNet
  40. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Technical University of Kosice, Kosice, Slovak Republic, 1999. View at MathSciNet
  41. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, vol. 31 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1957. View at MathSciNet
  42. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  43. S. D. Zaidman, Abstract Differential Equations, Pitman Advanced Publishing Program, San Francisco, Calif, USA, 1979. View at MathSciNet
  44. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44, Cambridge University Press, Cambridge, UK, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  45. Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1063–1077, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. A. Debbouche and M. M. El-Borai, “Weak almost periodic and optimal mild solutions of fractional evolution equations,” Electronic Journal of Differential Equations, no. 46, pp. 1–8, 2009. View at Zentralblatt MATH · View at MathSciNet
  47. M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433–440, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. Z. Yan, “Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay,” International Journal of Control, vol. 85, no. 8, pp. 1051–1062, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  49. Z. Yan, “Approximate controllability of fractional neutral integro-differental inclusions with state-dependent delay in Hilbert spaces,” IMA Journal of Mathematical Control and Information, 2012. View at Publisher · View at Google Scholar
  50. A. Debbouche and D. F. M. Torres, “Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces,” International Journal of Control, vol. 86, no. 9, pp. 1577–1585, 2013. View at Publisher · View at Google Scholar