This work focuses on linear finite-dimensional output feedback control of the Kuramoto–Sivashinsky equation (KSE) with periodic boundary conditions. Under the assumption that the linearization of the KSE around the zero solution is controllable and observable, linear finite-dimensional output feedback controllers are synthesized that achieve stabilization of the zero solution, for any value of ...

We consider the periodic initial value problem for the Kuramoto–Sivashinsky (KS) equation. We approximate the solution by discretizing in time by implicit–explicit BDF schemes and in space by a pseudo–spectral method. We present the results of various numerical experiments.

In this work, we study the one-dimensional stabilized Kuramoto Sivashinsky equation with additive uncorrelated stochastic noise. The Eckhaus stable band of the deterministic equation collapses to a narrow region near the center of the band. This is consistent with the behavior of the phase diffusion constants of these states. Some connections to the phenomenon of state selection in driven out o...

In this article, the robust Stackelberg controllability (RSC) problem is studied for a nonlinear fourth-order parabolic equation, namely Kuramoto–Sivashinsky equation. When three external sources are acting into system, RSC consists essentially in combining two subproblems: first one saddle point among sources. Such called “follower control” and its associated “disturbance signal.” This procedu...

We investigate the bifurcation structure of the Kuramoto-Sivashinsky equation with homogeneous Dirichlet boundary conditions. Using hidden symmetry principles, based on an extended problem with periodic boundary conditions and O(2) symmetry, we show that the zero solution exhibits two kinds of pitchfork bifurcations: one that breaks the reflection symmetry of the system with Dirichlet boundary ...

An investigation of interior crisis of high dimensions in an extended spatiotemporal system exemplified by the Kuramoto-Sivashinsky equation is reported. It is shown that unstable periodic orbits and their associated invariant manifolds in the Poincaré hyperplane can effectively characterize the global bifurcation dynamics of high-dimensional systems.