Theoretical and numerical indirect stabilization of coupled wave equations with a single time-delayed damping

TL;DR

This work addresses the exponential stabilization of a pair of strongly coupled wave equations with a single time-delayed damping on the first equation, expressed as $u_{tt}-\Delta u + b y_t + a[\mu_1 u_t + \mu_2 u_t(t-\tau)]=0$ and $y_{tt}-\Delta y - b u_t=0$. The authors establish well-posedness via semigroup theory by recasting the system as an abstract Cauchy problem with a delay variable, and prove exponential energy decay under the conditions $\mu_2<\mu_1$ and $\tau\mu_2<\xi<\tau(2\mu_1-\mu_2)$ together with a GCC-based observability inequality. A 1D finite-difference scheme is developed that preserves the continuous energy decay, including a delay-handling approach using a ghost time step, and a discrete energy is shown to dissipate under the same parameter constraints. Numerical simulations corroborate the theoretical results, showing exponential stabilization when $\mu_2<\mu_1$ and instability when $\mu_2\ge\mu_1$, thus validating both the analysis and the energy-preserving numerical method for delayed damping in strongly coupled hyperbolic systems.

Abstract

The focal point of this paper is to theoretically investigate and numerically validate the effect of time delay on the exponential stabilization of a class of coupled hyperbolic systems with delayed and non-delayed dampings. The class in question consists of two strongly coupled wave equations featuring a delayed and non-delayed damping terms on the first wave equation. Through a standard change of variables and semi-group theory, we address the well-posedness of the considered coupled system. Thereon, based on some observability inequalities, we derive sufficient conditions guaranteeing the exponential decay of a suitable energy. On the other hand, from the numerical point of view, we validate the theoretical results in $1D$ domains based on a suitable numerical approximation obtained through the Finite Difference Method. More precisely, we construct a discrete numerical scheme which preserves the energy decay property of its continuous counterpart. Our theoretical analysis and implementation of our developed numerical scheme assert the effect of the time-delayed damping on the exponential stability of strongly coupled wave equations.

Figures (4)

• Figure 1: A 3-d landscape of the dynamics of the coupled wave equation with time delay $\tau=2$, when $\mu_1=1$ and $\mu_2=0.5;$ (a) the component $u;$ (b) the component $y.$
• Figure 2: The energy, $E(t)$, of the coupled waves equations with time delay $\tau=2$, when $\mu_1=1$ versus time for various values of $\mu_2$.
• Figure 3: A 3-d landscape of the dynamics of the coupled wave equation with time delay $\tau=2$, when $\mu_1=1,$$\mu_2=1.2$ and $\xi=0;$ (a) the component $u;$ (b) the component $y$.
• Figure 4: The energy, $E(t)$, of the coupled waves equations with time delay $\tau=2$, versus time for $\mu_1=1,$$\mu_2=1.2$ and $\xi=0$.

Theorems & Definitions (7)

• Theorem 2.1
• proof
• Lemma 3.1
• proof
• Theorem 3.1
• Theorem 3.2
• proof