Feedback stabilization of some fourth-order nonlinear parabolic equations with saturated controls

TL;DR

This paper develops a stabilization method for certain nonlinear parabolic equations using saturated feedback control, spectral analysis, and modal decomposition, achieving local exponential stability in a specific function space.

Contribution

It introduces a novel stabilization strategy for fourth-order nonlinear parabolic equations with saturated controls, combining spectral analysis and LMIs.

Findings

Achieved local exponential stabilization in H^2 space.

Identified finite unstable eigenvalues for the system.

Designed stabilization using modal decomposition and geometric conditions.

Abstract

In this work, we analyze the internal and boundary stabilization of the Cahn-Hilliard and Kuramoto-Sivashinsky equations under saturated feedback control. We conduct our study through the spectral analysis of the associated linear operator. We identify a finite number of eigenvalues related to the unstable part of the system and then design a stabilization strategy based on modal decomposition, linear matrix inequalities (LMIs), and geometric conditions on the saturation function. Local exponential stabilization in $H^{2}$ is established.