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Stabilization of a Wave-Heat Cascade System

Hugo Lhachemi, Christophe Prieur, Emmanuel Trélat

TL;DR

This work tackles the problem of stabilizing a one-dimensional wave–heat cascade with internal coupling, using an explicit output-feedback law based on a finite number of modal components. By exploiting a Riesz-spectral structure, the authors perform spectral reduction to isolate unstable parabolic modes and design a finite-dimensional observer plus state-feedback that achieves exponential decay in both the energy space and a stronger parabolic norm, with a prescribed rate. The stabilization remains valid under explicit controllability and observability conditions and extends to pointwise measurements, aided by a Lyapunov functional built on Riesz bases. The results offer a principled, implementable approach to boundary control of coupled hyperbolic–parabolic PDEs, with demonstrated numerical support and clear directions for broader applicability.

Abstract

We consider the output-feedback stabilization of a one-dimensional cascade coupling a reaction-diffusion equation and a wave equation through an internal term, with Neumann boundary control acting at the wave endpoint. Two measurements are available: the wave velocity at the controlled boundary and a temperature-type observation of the reaction-diffusion component, either distributed or pointwise. Under explicit, necessary and sufficient conditions on the coupling and observation profiles, we show that the generator of the open-loop system is a Riesz-spectral operator. Exploiting this structure, we design a finite-dimensional dynamic output-feedback law, based on a finite number of parabolic modes, which achieves arbitrary exponential decay in both the natural energy space and a stronger parabolic norm. The construction relies on a spectral reduction and a Lyapunov argument in Riesz bases. We also extend the design to pointwise temperature or heat-flux measurements.

Stabilization of a Wave-Heat Cascade System

TL;DR

This work tackles the problem of stabilizing a one-dimensional wave–heat cascade with internal coupling, using an explicit output-feedback law based on a finite number of modal components. By exploiting a Riesz-spectral structure, the authors perform spectral reduction to isolate unstable parabolic modes and design a finite-dimensional observer plus state-feedback that achieves exponential decay in both the energy space and a stronger parabolic norm, with a prescribed rate. The stabilization remains valid under explicit controllability and observability conditions and extends to pointwise measurements, aided by a Lyapunov functional built on Riesz bases. The results offer a principled, implementable approach to boundary control of coupled hyperbolic–parabolic PDEs, with demonstrated numerical support and clear directions for broader applicability.

Abstract

We consider the output-feedback stabilization of a one-dimensional cascade coupling a reaction-diffusion equation and a wave equation through an internal term, with Neumann boundary control acting at the wave endpoint. Two measurements are available: the wave velocity at the controlled boundary and a temperature-type observation of the reaction-diffusion component, either distributed or pointwise. Under explicit, necessary and sufficient conditions on the coupling and observation profiles, we show that the generator of the open-loop system is a Riesz-spectral operator. Exploiting this structure, we design a finite-dimensional dynamic output-feedback law, based on a finite number of parabolic modes, which achieves arbitrary exponential decay in both the natural energy space and a stronger parabolic norm. The construction relies on a spectral reduction and a Lyapunov argument in Riesz bases. We also extend the design to pointwise temperature or heat-flux measurements.
Paper Structure (21 sections, 97 equations, 2 figures)

This paper contains 21 sections, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Studied problem
  3. State-of-the-art and contribution
  4. Paper organization and technical outline
  5. Preliminary velocity feedback and spectral properties
  6. Preliminary velocity feedback
  7. Spectral properties of $\mathcal{A}$
  8. Spectral reduction
  9. Homogeneous representation and spectral reduction
  10. Finite-dimensional model and controllability properties
  11. Finite-dimensional model
  12. Controllability
  13. Discussion on the coefficients $\gamma_n$
  14. Measurement
  15. Feedback stabilization
  16. ...and 6 more sections

Figures (2)

  • Figure 1: $\gamma_2$ as a function of $b \in [0,1]$, with $L=1$, $c=50$, $\beta_0 = 1$, and $a=0$.
  • Figure 2: Trajectory of the wave-heat cascade \ref{['eq: cascade equation']} in closed loop

Theorems & Definitions (8)

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