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Exponential stability for the generalized Korteweg-de Vries equation in a finite interval with weak damping. (English) Zbl 1421.35320

Summary: The aim of this paper is to consider the generalized Korteweg-de Vries equation in a finite interval with a very weak localized dissipation. We obtain the globally uniformly exponentially stability of this equation. The main difficulty in this context comes from the structure of nonlinear term and the lack of regularity.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
93C20 Control/observation systems governed by partial differential equations
93D15 Stabilization of systems by feedback
35B35 Stability in context of PDEs
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[1] E. Cerpa and E. Crépeau, Rapid exponential stabilization for a linear Korteweg-de Vries equation, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 655-668. · Zbl 1161.93018 · doi:10.3934/dcdsb.2009.11.655
[2] E. Cerpa and J. M. Coron, Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Trans. on Autom. Control, 58 (2013), 1688-1695. · Zbl 1369.93480 · doi:10.1109/TAC.2013.2241479
[3] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443. · JFM 26.0881.02 · doi:10.1080/14786449508620739
[4] C. P. Massarolo, G. P. Menzala, and A. F. Pazoto, On the uniform decay for the Korteweg-de Vries equation with weak damping, Math. Methods Appl. Sci., 12 (2007), 1419-1435. · Zbl 1114.93080 · doi:10.1002/mma.847
[5] R. M. Miura, The Korteweg-de Vries equation: A survey of results, SIAM Rev., 18 (1976), 412-459. · Zbl 0333.35021 · doi:10.1137/1018076
[6] A. F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486. · Zbl 1148.35348 · doi:10.1051/cocv:2005015
[7] G. Perla Menzala, C. F. Vasconcellos, and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129. · Zbl 1039.35107 · doi:10.1090/qam/1878262
[8] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55. · Zbl 0873.93008 · doi:10.1051/cocv:1997102
[9] L. Rosier and B-Y Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45(3) (2006), 927-956. · Zbl 1116.35108 · doi:10.1137/050631409
[10] D. L. Russell and B.-Y. Zhang, Smoothing properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation, J. Math. Anal. Appl., 190 (1995), 449-488. · Zbl 0845.35111 · doi:10.1006/jmaa.1995.1087
[11] S. M. Sun, The Korteweg-de Vries equation on a periodic domain with singular-point dissipation, SIAM J. Control Optim., 34 (1996), 892-912. · Zbl 0937.93023 · doi:10.1137/S0363012994269491
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