Boundary control of the Kuramoto-Sivashinsky equation with low anti-dissipation

Abstract

We address the problem of Dirichlet and Neumann boundary control of the Kuramoto-Sivashinsky equation on the domain [0, 1]. We note that, while the uncontrolled Dirichlet problem is asymptotically stable when an "anti-diffusion" parameter is small, and unstable when it is large (the critical value of the parameter), the uncontrolled Neumann problem is never asymptotically stable. We develop a Neumann feedback law that guarantees L/sup 2/-global exponential stability and H/sup 2/-global asymptotic stability for small values of the anti-diffusion parameter. The more interesting problem of boundary stabilization when the anti-diffusion parameter is large remains open. Our proof of global existence and uniqueness of solutions of the closed-loop system involves construction of a Green function and application of the Banach contraction mapping principle.