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On feedback stabilisation for the Cahn-Hilliard equation and its numerical approximation

Herbert Egger, Marvin Fritz, Karl Kunisch, Sérgio S. Rodrigues

TL;DR

This work addresses stabilising the Cahn-Hilliard equation toward a given target trajectory using finite-dimensional static output feedback. It derives continuous-time exponential stabilisation via energy estimates under a spectral condition on the feedback operator and shows that the results extend to discretisations, with explicit constructions of pointwise and averaged actuators that satisfy the spectral inequality. The paper also provides a fully discrete stability theory and numerical validation, demonstrating calibration between actuator density and feedback gain and verifying discretisation-independence of the stabilisation behavior. The findings have practical implications for trajectory tracking in phase-field models and support robust numerical schemes for controlled Cahn-Hilliard dynamics.

Abstract

We consider the stabilisation of solutions to the Cahn-Hilliard equation towards a given trajectory by means of a finite-dimensional static output feedback mechanism. Exponential stabilisation of the controlled state around the target trajectory is proven using careful energy estimates and a spectral condition which characterizes the strength of the feedback. The analysis is general enough to allow for pointwise and distributed measurements and actuation. The main results are derived via arguments that carry over to appropriate discretisation schemes which allows us to establish corresponding exponential stabilisation results also on the discrete level. The validity of our results and the importance of some of our assumptions are illustrated by numerical tests.

On feedback stabilisation for the Cahn-Hilliard equation and its numerical approximation

TL;DR

This work addresses stabilising the Cahn-Hilliard equation toward a given target trajectory using finite-dimensional static output feedback. It derives continuous-time exponential stabilisation via energy estimates under a spectral condition on the feedback operator and shows that the results extend to discretisations, with explicit constructions of pointwise and averaged actuators that satisfy the spectral inequality. The paper also provides a fully discrete stability theory and numerical validation, demonstrating calibration between actuator density and feedback gain and verifying discretisation-independence of the stabilisation behavior. The findings have practical implications for trajectory tracking in phase-field models and support robust numerical schemes for controlled Cahn-Hilliard dynamics.

Abstract

We consider the stabilisation of solutions to the Cahn-Hilliard equation towards a given trajectory by means of a finite-dimensional static output feedback mechanism. Exponential stabilisation of the controlled state around the target trajectory is proven using careful energy estimates and a spectral condition which characterizes the strength of the feedback. The analysis is general enough to allow for pointwise and distributed measurements and actuation. The main results are derived via arguments that carry over to appropriate discretisation schemes which allows us to establish corresponding exponential stabilisation results also on the discrete level. The validity of our results and the importance of some of our assumptions are illustrated by numerical tests.
Paper Structure (20 sections, 6 theorems, 52 equations, 4 figures)

This paper contains 20 sections, 6 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Problem setting
  4. Notation and main result
  5. Outline of the manuscript
  6. Proof of Theorem \ref{['thm:1']}
  7. Weak formulation
  8. Galerkin approximation
  9. Passage to the limit
  10. Proof of Theorem \ref{['thm:1']}.
  11. Uniqueness
  12. Construction of feedback operators
  13. Pointwise control
  14. Further constructions and monotonicity
  15. Discretisation
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let Assumption ass:1 hold with $\nu,R>0$. Then for any $T>0$ and any choice of $y_0\in L^2(\Omega)$, $h \in L^2(0,T;L^2)$, problem eq:ch1--eq:ch2 has a weak solution satisfying the initial condition $y(0)=y_0$ and the estimate for all $0 \le t \le T$ and any $\gamma$ satisfying condition eq:spectral. Solutions thus exist globally in time and, if $\gamma > 0$ and $h=h_r$, they stabilise exponenti

Figures (4)

  • Figure 1: Arrangement of feedback measurements and actuators for a domain $\Omega=(0,1)^2$ with $M=1,2,4$ subintervals per coordinate direction. Red dots correspond to pointwise actuators and measurements whereas green areas depict corresponding characteristic functions.
  • Figure 2: Contour plots of the minimal eigenvalue $\alpha_{\min}(M, \lambda)$ for two spatial mesh sizes $h$ (solid lines) and $h/2$ (dashed lines), with parameter $\nu \in\{0.01,0.001\}$; the red dotted line indicates the theoretical threshold value $C^*$ for stabilisation where we chose $R=1$.
  • Figure 3: Snapshots of of solutions $y(t,x)$ to controlled system \ref{['eq:ch1']}--\ref{['eq:ch2']} with pointwise feedback $\mathcal{F}=\mathcal{F}_{M,\lambda}$ for time $t \in \{0,0.005,0.05,0.1,1\}$ (left to right) and different combinations of $M$ and $\lambda$ (top to bottom).
  • Figure 4: Evolution of distance $\log\|z(t)\|_{L^2}^2$ for various combinations of $M$ and $\lambda$; left plot: Non-convergence for too small $\lambda$ or $M$; right plot: exponential decay for $M$ and $\lambda$ sufficiently large.

Theorems & Definitions (17)

  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • ...and 7 more