Rapid Fredholm stabilization of the Kuramoto--Sivashinsky equation with unrestricted, spatially-varying anti-diffusion

Luke Bhan; Miroslav Krstic; Yuanyuan Shi

eess.SYOct 6, 2026

Rapid Fredholm stabilization of the Kuramoto--Sivashinsky equation with unrestricted, spatially-varying anti-diffusion

Authors: Luke Bhan, Miroslav Krstic, Yuanyuan Shi

Organizations: University of California, San Diego, 3180 Voigt Drive, La Jolla, CA 92093, USA.

Abstract

We develop 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 a loss of controllability. We overcome this obstruction by introducing a second boundary input and assigning the two inputs distinct roles. The key idea, inspired by Heymann's Lemma, is to use the boundary value $u(0,t)$ entirely for a pre-feedback that renders the modified plant controllable through the curvature input $u_{xx}(0,t)$ . The latter input then stabilizes the plant through a Fredholm backstepping transformation. We show that two inputs suffice for controllability and are necessary when the plant has an unstable double eigenvalue. However, the Fredholm kernel still must be approximated for implementation. Hence, to enable kernel and gain approximation, we prove continuity of the coefficient-to-gain design map on compact admissible design classes. Unlike Volterra-based continuity proofs using successive approximations, our proof uses the modal representation to control the spectral data, the inverse coefficient system, and the tails of the kernel and gain series. This yields a single neural operator approximation of the gain to any prescribed $L^2$ accuracy across the class. Finally, we establish rapid local stabilization of the nonlinear closed-loop system under both the exact gains and sufficiently accurate approximations. We conclude with numerical results that illustrate prescribed decay rates and the computational cost of the approximations. In particular, we train a Fourier neural operator that achieves typical relative gain errors of approximately $0.1\%$ and stabilizes all held-out cases tested, including a plant with an unstable double eigenvalue.

Figures & tables

Result	Statement	Section
Theorem 2.3 , Proposition 2.3	Two inputs suffice for every coefficient. A nonnegative double eigenvalue obstructs stabilization through either input alone.	2
Theorem 4.1	Pre-feedback ensures simple spectrum and controllability through the curvature input.	4
Theorem 5.1	The kernel and gain exist, with explicit modal formulas and an invertible Fredholm transformation.	5
Theorem 6.2	The kernel and gain depend continuously on a compact admissible design class, with uniformly bounded design and stability constants.	6
Theorem 6.4	One neural operator approximates the gain uniformly to any prescribed accuracy.	6.4
Theorem 1.1 , Theorem 7.2	Exact feedback achieves local exponential stability at any prescribed decay rate.	7

Table 1. Main results and their roles in the design.

Panel of Figure 1 — Figure 1. Two feedbacks with distinct roles. (a) Displacement pre-feedback (blue) makes every mode controllable from the curvature input. The Fredholm design then constructs the stabilizing gain $h$ and transformation $T=I-K$ . (b) Both feedback laws act simultaneously on the same plant state, with $g_{\theta}(y)=e^{-\theta y}$ . The shift $a$ sets the achievable decay rate.

Figure 2. Stabilization of the nonlinear plant with coefficient ( 8.1 ), initial condition $u_{0}(x)=10^{-3}\sqrt{2}\sin(\pi x)$ , and $(a,\theta,\varepsilon)=(480,1,0.5)$ . (a) The coefficient $\lambda(x)$ . (b) The Fredholm kernel computed with $32$ modes. (c) The two boundary controls. (d) The nonlinear plant state.

Figure 3. Selectable decay rates for the fixed plant ( 8.1 ), with

(\theta,\varepsilon)=(1,0.5)

and

u_{0}(x)=10^{-3}\sqrt{2}\sin(\pi x)

. (a) Nonlinear state norms for

a=480,600,1000

. (b) The corresponding curvature controls, with the first

0.03

seconds displayed to resolve the transient. Each curvature gain is recomputed using

1536

modes.

Figure 5. Gain approximation and nonlinear stabilization for the coefficient ( 8.1 ), with $(a,\theta,\varepsilon)=(1000,1,4)$ and $u_{0}(x)=10^{-3}\sqrt{2}\sin(\pi x)$ . (a) Gains from $N=4,1024$ and FNO. (b) Relative pointwise error $|\hat{h}(y)-h_{\rm ref}(y)|/(|h_{\rm ref}(y)|+10^{-6})$ , expressed as a percentage. The FNO relative $L^{2}$ error is $0.1107\%$ . (c) FNO controls $U_{1}$ (blue, left axis) and $U_{2}$ (orange, right axis). (d) Nonlinear state under neural feedback up to $t=0.015$ .

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