Numerical stabilization method by switching time-delay

TL;DR

This work addresses the stabilization of evolution systems with time-delayed damping by employing switching time-delay, extending the Ammari–Nicaise–Pignotti framework to both analysis and numerical approximation. It provides well-posedness results via semigroup theory, exponential stability for a shifted generator under a key parameter condition, and develops 1D numerical schemes (finite difference and finite volume) that respect discrete energy and CFL constraints. The paper validates theory through comprehensive 1D experiments across boundary, internal, and pointwise delay setups, identifying critical damping levels where energy transitions from decay to growth. The findings offer a practical approach to time-delay compensation in PDEs and pave the way for extending energy-stable schemes to more complex, partially damped, or coupled systems.

Abstract

In this paper, we propose a new numerical strategy for the stabilization of evolution systems. The method is based on the methodology given by Ammari, Nicaise andPignotti in ''Stabilization by switching time-delay, Asymptot. Anal., 83 (2013), 263--283''. This method is then implemented in 1D by suitable numerical approximation techniques. Numerical experiments complete this study to confirm the theoretical announced results.

Figures (19)

• Figure 1: A model representing the admissible one-dimensional mesh
• Figure 2: Boundary delayed control. Energy when $0 < \mu < 1$
• Figure 3: Boundary delayed control. Exponential decay $0 < \mu < 1$
• Figure 4: Boundary delayed control. Exponential decay rate $0 < \mu < 1$
• Figure 5: Boundary delayed control. The surprising case $\mu = 1$

Theorems & Definitions (29)

• Lemma 2.1
• proof
• Proposition 2.2
• Remark 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• Theorem 3.1
• proof : Proof of Theorem \ref{['lr']}