Distributed control of the Kuramoto-Sivashinsky equation using approximations

Abstract

The Kuramoto-Sivashinsky (KS) equation is a non-linear partial differential equation first derived for the study of chemical reaction systems. For some parameter values, this equation is unstable. We consider bounded control of the KS equation with a single control. It is shown that stabilizing the linearized KS equation implies local exponential stability of the nonlinear controlled system. This is used to develop a strategy for controller design using a lumped approximation. An example is presented to illustrate the approach. These results indicate the system is stabilized and that spillover is avoided. The numerical results also indicate that the approach can also be used to steer the system from one equilibrium point to another.