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

TL;DR

This work develops a spectral-decomposition framework for stabilizing fourth-order nonlinear parabolic equations (CH and KS) under input saturation. By isolating a finite set of unstable modes and stabilizing them with LMIs under a saturating feedback, the authors prove local exponential stabilization in $L^{2}$ and $H^{2}$, and extend the results to fully nonlinear regimes. A cascade argument shows that stabilizing the finite-dimensional unstable subspace suffices for the whole system, with careful nonlinear estimates ensuring global well-posedness under small data. The approach also covers boundary control and highlights potential extensions using admissible-operator theory for analytic semigroups in higher dimensions. Overall, the paper provides a constructive methodology for saturated stabilization of high-order parabolic PDEs with practical implications for actuation constraints.

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.

Theorems & Definitions (41)

• Remark 2.1
• Lemma 2.1
• Remark 3.1
• Remark 3.2
• Definition 3.1: Region of attraction
• Definition 3.2
• Theorem 3.1
• Proposition 3.1
• Proposition 3.2
• proof