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Exact null controllability of KdV-Burgers equation with memory effect systems. (English) Zbl 1256.49008
Summary: This paper is concerned with exact null controllability analysis of the nonlinear KdV-Burgers equation with memory. The proposed approach relies upon regression tools to prove the controllability property of the linearized KdV-Burgers equation via Carleman estimates. The control is distributed along with subdomain $\omega \subset \Omega$ and the external control acts on the key role of observability inequality with memory. This description finally shows the exact null controllability guaranteeing the stability.

MSC:
 49J20 Optimal control problems with PDE (existence) 35Q53 KdV-like (Korteweg-de Vries) equations 93B05 Controllability 93D20 Asymptotic stability of control systems
Full Text:
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. · Zbl 0860.76002 [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. · Zbl 0805.60053 · doi:10.1214/aoap/1177004974 [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. · Zbl 0818.60046 · doi:10.1007/BF02099589 [4] S. Kida, “Asymptotic properties of Burgers turbulence,” Journal of Fluid Mechanics, vol. 93, no. 2, pp. 337-377, 1979. · Zbl 0436.76031 · doi:10.1017/S0022112079001932 [5] J. M. Burgers, The Nonlinear Diffusion Equations, Reidel, Dordrecht, The Netherlands, 1974. · Zbl 0302.60048 [6] O. V. Rudenkov and S. I. Soluyan, Theoretical Foundation of Nonlinear Acoustics, New york, NY, USA, 1977. · Zbl 0413.76059 [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. [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. · Zbl 1106.93009 · doi:10.1051/cocv:2006010 · numdam:COCV_2006__12_3_442_0 · eudml:245950 [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. · Zbl 0990.93008 [10] V. Barbu, “Controllability of parabolic and Navier-Stokes equations,” Scientiae Mathematicae Japonicae, vol. 56, no. 1, pp. 143-211, 2002. · Zbl 1010.93054 [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. · Zbl 1007.93034 [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. · Zbl 0862.49004 [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. · Zbl 0966.93055 · doi:10.1137/S0363012999353229 [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. · Zbl 0862.93035 · doi:10.1090/S0002-9947-96-01672-8 [15] R. Sakthivel, “Robust stabilization the Korteweg-de Vries-Burgers equation by boundary control,” Nonlinear Dynamics, vol. 58, no. 4, pp. 739-744, 2009. · Zbl 1183.76660 · doi:10.1007/s11071-009-9514-z [16] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, vol. 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1983. · Zbl 0522.35002 [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. · Zbl 0760.93067 · doi:10.1007/BF01211563 [18] R. A. Adams, Sobolev Spaces, Academic Press, New York, NY, USA, 2nd edition, 2003. · Zbl 0527.55016