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Global boundary stabilization of the Korteweg-de Vries-Burgers equation. (English) Zbl 1125.35404

Summary: The problem of global exponential stabilization by boundary feedback for the Korteweg-de Vries-Burgers equation on the domain \([0,1]\) is considered. We derive a control law of the form \(u(0)=u_x(1)= u_{xx}(1)-k[u(1)^3+u(1)]=0\), where \(k\) is a sufficiently large positive constant, and prove that it guarantees \(L^2\)-global exponential stability, \(H^3\)-global asymptotic stability, and \(H^3\)-semiglobal exponential stability. Our decay rate estimates depend not only on the diffusion coefficient but also on the dispersion coefficient. The closed-loop system is shown to be well-posed.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
93D15 Stabilization of systems by feedback
35B35 Stability in context of PDEs