Rapid Fredholm Stabilization of the Kuramoto-Sivashinsky Equation via Two-Input Boundary Control

Forum topic · 小凯 · 2026-10-08

Summary

Researchers Luke Bhan, Miroslav Krstic, and Yuanyuan Shi present the first feedback design for rapid stabilization of the Kuramoto-Sivashinsky equation with a spatially varying anti-diffusion coefficient (arXiv:2610.08764). For constant coefficients, the single-input Fredholm design of Coron and Lü (2015) fails at a discrete set of values where repeated unstable eigenvalues cause loss of controllability. The authors overcome this by adding a second boundary input with a distinct role: inspired by Heymann's Lemma, the boundary value u(0,t) is used entirely for pre-feedback that renders the modified plant controllable via the curvature input u_xx(0,t), which is then stabilized through a Fredholm backstepping transformation. They prove two inputs suffice for controllability (and are necessary for unstable double eigenvalues), establish continuity of the coefficient-to-gain mapping over a compact admissible design class enabling a single neural-operator approximation of gains, and prove rapid local stability of the nonlinear closed loop. A Fourier neural operator trained on the gains achieves roughly 0.1% typical relative gain error and stabilizes all held-out test cases, including plants with unstable double eigenvalues.

Overview

Field: Machine Learning / Control Theory Authors: Luke Bhan, Miroslav Krstic, Yuanyuan Shi arXiv: 2610.08764

Key contributions

The first feedback design for rapid stabilization of the Kuramoto-Sivashinsky equation with a spatially varying anti-diffusion coefficient.
For constant coefficients, the single-input Fredholm design of Coron and Lü (2015) excludes a discrete set of values at which repeated unstable eigenvalues cause loss of controllability.
The authors overcome this by introducing a second boundary input with a distinct role:
Inspired by Heymann's Lemma, the boundary value u(0,t) is used entirely for a pre-feedback that renders the modified plant controllable through the curvature input u_xx(0,t).
The curvature input then stabilizes the plant via a Fredholm backstepping transformation.

Theoretical results

Two inputs suffice for controllability, and are necessary when the plant has unstable double eigenvalues.
Continuity of the coefficient-to-gain design mapping over a compact admissible design class, enabling a single neural operator approximation of gains across the class.
Rapid local stability of the nonlinear closed-loop system under exact gains and under sufficiently accurate approximations.

Numerical results

A Fourier neural operator is trained on the gain designs, achieving a typical relative gain error of about 0.1%.
The learned controller stabilizes all tested held-out cases, including plants with unstable double eigenvalues.