Output feedback control of the Kuramoto-Sivashinsky equation

The Kuramoto-Sivashinsky equation is a nonlinear partial differential equation that models reaction-diffusion systems. The stability of the equilibria depends on the value of a positive parameter; the set of all constant equilibria are unstable when the instability parameter is less than 1. Stabilization of the Kuramoto-Sivashinsky equation using scalar output-feedback control is considered in this paper. This is done by stabilizing the corresponding linearized system. A finite-dimensional controller is then designed to stabilize the system. Fréchet differentiability of the semigroup generated by the closed-loop system plays an important role in proving that this approach yields a locally stable equilibrium. The approach is illustrated with a numerical example.