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 |