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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 531659, 20 pages
http://dx.doi.org/10.1155/2012/531659
Research Article

Exact Null Controllability of KdV-Burgers Equation with Memory Effect Systems

1CASY, DEIS, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy
2Advanced Electronics and Information Research Center, Department of Electronics and Information Technology, Chonbuk National University, Republic of Korea
3Department of Electrical and Electronics Engineering, Seonam University, Republic of Korea

Received 12 August 2012; Accepted 7 November 2012

Academic Editor: Yongfu Su

Copyright © 2012 Rajagounder Ravi Kumar 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. S. N. Gurbatov, A. N. Malakhov, and A. I. Saichev, Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays, Particles, Manchester University Press, Manchester, UK, 1991.
  2. Y. Hu and W. A. Woyczyński, “An extremal rearrangement property of statistical solutions of Burgers' equation,” The Annals of Applied Probability, vol. 4, no. 3, pp. 838–858, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. A. Molchanov, D. Surgailis, and W. A. Woyczyński, “Hyperbolic asymptotics in Burgers' turbulence and extremal processes,” Communications in Mathematical Physics, vol. 168, no. 1, pp. 209–226, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. S. Kida, “Asymptotic properties of Burgers turbulence,” Journal of Fluid Mechanics, vol. 93, no. 2, pp. 337–377, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J. M. Burgers, The Nonlinear Diffusion Equations, Reidel, Dordrecht, The Netherlands, 1974.
  6. O. V. Rudenkov and S. I. Soluyan, Theoretical Foundation of Nonlinear Acoustics, New york, NY, USA, 1977.
  7. L. Kofman, D. Pogosyan, S. F. Shandarin, and A. L. Melott, “Coherent structures in the universe and the adhesion model,” Astrophysical Journal Letters, vol. 393, no. 2, pp. 437–449, 1992. View at Scopus
  8. E. Fernández-Cara, M. González-Burgos, S. Guerrero, and J.-P. Puel, “Null controllability of the heat equation with boundary Fourier conditions: the linear case,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 12, no. 3, pp. 442–465, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. V. Barbu and M. Iannelli, “Controllability of the heat equation with memory,” Differential and Integral Equations, vol. 13, no. 10–12, pp. 1393–1412, 2000. View at Zentralblatt MATH
  10. V. Barbu, “Controllability of parabolic and Navier-Stokes equations,” Scientiae Mathematicae Japonicae, vol. 56, no. 1, pp. 143–211, 2002. View at Zentralblatt MATH
  11. E. Fernández-Cara and E. Zuazua, “The cost of approximate controllability for heat equations: the linear case,” Advances in Differential Equations, vol. 5, no. 4–6, pp. 465–514, 2000. View at Zentralblatt MATH
  12. A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, vol. 34, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, Korea, 1996.
  13. L. Rosier, “Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line,” SIAM Journal on Control and Optimization, vol. 39, no. 2, pp. 331–351, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. D. L. Russell and B. Y. Zhang, “Exact controllability and stabilizability of the Korteweg-de Vries equation,” Transactions of the American Mathematical Society, vol. 348, no. 9, pp. 3643–3672, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. R. Sakthivel, “Robust stabilization the Korteweg-de Vries-Burgers equation by boundary control,” Nonlinear Dynamics, vol. 58, no. 4, pp. 739–744, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, vol. 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1983.
  17. J.-M. Coron, “Global asymptotic stabilization for controllable systems without drift,” Mathematics of Control, Signals, and Systems, vol. 5, no. 3, pp. 295–312, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. R. A. Adams, Sobolev Spaces, Academic Press, New York, NY, USA, 2nd edition, 2003. View at Zentralblatt MATH