A new theorem is applied to the Kuramoto-Sivashinsky equation with L-periodic boundary conditions, proving the existence of an asymptotically complete inertial manifold attracting all initial data. Combining this result with a new estimate of the size of the globally absorbing set yields an improved estimate of the dimension, N ∼ L.

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 present the Girsanov theorem for a non linear Itô equation in an infinite dimensional Hilbert space with a non linearity of polynomial growth and an infinite dimensional additive noise. We assume a condition weaker than Novikov one, as done by Mikulevicius and Rozovskii in the study of more general stochastic PDE’s. The equivalence of the laws of the linear equation and of the non linear equ...

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 report numerical simulations of one-dimensional cellular solutions of the stabilized Kuramoto-Sivashinsky equation. This equation offers a range of generic behavior in pattern-forming instabilities of moving interfaces, such as a host of secondary instabilities or transition toward disorder. We compare some of these collective behaviors to those observed in experiments. In particular, destab...

The (Michelson) Sivashinsky equation of premixed flames is studied in a rectangular domain in two dimensions. A huge number of two-dimensional (2D) stationary solutions are trivially obtained by the addition of two 1D solutions. With Neumann boundary conditions, it is shown numerically that adding two stable 1D solutions leads to a 2D stable solution. This type of solution is shown to play an i...