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Stability of abstract coupled systems

Serge Nicaise, Lassi Paunonen, David Seifert

TL;DR

The paper develops an abstract two-component framework for boundary-coupled linear evolution equations using impedance-passive boundary nodes and transfer functions to derive resolvent estimates and decay rates. By reducing the coupled dynamics to the dissipative part and a boundary transfer, it yields explicit resolvent bounds and translates them into decay rates, including strong, exponential, and polynomial stability, for PDE models such as a one-dimensional wave-heat system, wave-heat networks, and a wave equation with an acoustic boundary condition. The approach relies on identities like $L H(\lambda)=\lambda H(\lambda)$, $G H(\lambda)=I$, $K H(\lambda)=P(\lambda)$ and the impedance-passivity of the boundary nodes to enable a divide-and-conquer analysis. The results provide a unified and practical toolkit for energy-decay estimates in boundary-coupled PDEs with mixed hyperbolic-parabolic structure.

Abstract

We study stability of abstract differential equations coupled by means of a general algebraic condition. Our approach is based on techniques from operator theory and systems theory, and it allows us to study coupled systems by exploiting properties of the components, which are typically much simpler to analyse. As our main results we establish resolvent estimates and decay rates for abstract boundary-coupled systems. We illustrate the power of the general results by using them to obtain rates of energy decay in coupled systems of one-dimensional wave and heat equations, and in a wave equation with an acoustic boundary condition.

Stability of abstract coupled systems

TL;DR

The paper develops an abstract two-component framework for boundary-coupled linear evolution equations using impedance-passive boundary nodes and transfer functions to derive resolvent estimates and decay rates. By reducing the coupled dynamics to the dissipative part and a boundary transfer, it yields explicit resolvent bounds and translates them into decay rates, including strong, exponential, and polynomial stability, for PDE models such as a one-dimensional wave-heat system, wave-heat networks, and a wave equation with an acoustic boundary condition. The approach relies on identities like , , and the impedance-passivity of the boundary nodes to enable a divide-and-conquer analysis. The results provide a unified and practical toolkit for energy-decay estimates in boundary-coupled PDEs with mixed hyperbolic-parabolic structure.

Abstract

We study stability of abstract differential equations coupled by means of a general algebraic condition. Our approach is based on techniques from operator theory and systems theory, and it allows us to study coupled systems by exploiting properties of the components, which are typically much simpler to analyse. As our main results we establish resolvent estimates and decay rates for abstract boundary-coupled systems. We illustrate the power of the general results by using them to obtain rates of energy decay in coupled systems of one-dimensional wave and heat equations, and in a wave equation with an acoustic boundary condition.
Paper Structure (7 sections, 17 theorems, 122 equations)

This paper contains 7 sections, 17 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminary results on boundary nodes
  3. Abstract coupled systems
  4. Stability of coupled PDE models
  5. A one-dimensional wave-heat system
  6. Wave-heat networks
  7. A wave equation with an acoustic boundary condition

Key Result

Theorem 1.1

Let $(G_1, L_1, K_1)$ and $(G_2,L_2, K_2)$ be two impedance passive boundary nodes and let $A_0$ denote the restriction of $L_1$ to $\mathop{\mathrm{Ker}}\limits (G_1+K_1)$. Suppose that $i\mathbb{R}\subseteq \rho(A_0)\cap \rho(A_2)$ and let $M_0\colon \mathbb{R}\to [r,\infty)$, for some $r>0$, be Suppose further that there exists a function $\eta\colon \mathbb{R}\to(0,\infty)$ such that and th

Theorems & Definitions (44)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • ...and 34 more