The Stability of a Coupled Degenerate Wave System Under Boundary Control

1. Problem Statement

The paper investigates the asymptotic stability of a coupled system consisting of two degenerate wave equations defined on adjacent intervals:

• Right String ($x \in (0, 1)$): Governed by $u_{tt} - (a(x)u_x)_x = 0$, where the diffusion coefficient $a(x) = x^{\mu_a}$ degenerates at $x=0$ (with $0 \le \mu_a < 1$).
• Left String ($x \in (-1, 0)$): Governed by $w_{tt} - (b(x)w_x)_x = 0$, where $b(x) = (x+1)^{\mu_b}$ degenerates at $x=-1$ (with $0 \le \mu_b < 2$).

Coupling and Control:

• The two strings are connected at the interface $x=0$ via continuity of displacement ($u=w$) and continuity of flux ($\lim_{x\to 0^+} a u_x = b w_x$).
• Boundary Control: A feedback control is applied only at the free endpoint of the right string ($x=1$) via the condition $\gamma u(1,t) + u_t(1,t) + u_x(1,t) = 0$ (where $\gamma > 0$).
• The left endpoint ($x=-1$) is either fixed ($w=0$ if $\mu_b < 1$) or has a zero flux condition (if $\mu_b \ge 1$).

The primary objective is to determine the decay rate of the system's energy $E(t)$ under this single boundary control, specifically analyzing how the degree of degeneracy ($\mu_a$ and $\mu_b$) influences stability.

2. Methodology

The authors employ a rigorous functional analytic approach combined with the frequency domain method.

• Functional Setting:

  • The system is formulated in appropriate weighted Sobolev spaces ($V^1_\alpha$ and $V^2_\alpha$) tailored to the degeneracy of the coefficients.
  • The problem is rewritten as an abstract evolution equation $\frac{d}{dt}U(t) = \mathcal{A}_i U(t)$ in a Hilbert energy space $\mathcal{H}_i$.
  • Two cases are considered based on the degeneracy of the left string ($\mu_b < 1$ vs. $\mu_b \ge 1$), leading to different domain definitions for the operator $\mathcal{A}$.

• Well-posedness:

  • Using the Lumer-Phillips theorem, the authors prove that the operator $\mathcal{A}$ generates a $C_0$-semigroup of contractions.
  • They establish that the resolvent set contains the imaginary axis ($i\mathbb{R} \subset \rho(\mathcal{A})$), ensuring the system does not have purely imaginary eigenvalues (which would imply non-decay).

• Stability Analysis (Frequency Domain Method):

  • Instead of constructing a Lyapunov functional, the authors utilize a characterization of polynomial stability for $C_0$-semigroups (Lemma 3.3).
  • Key Criterion: The system is polynomially stable with order $\theta$ if and only if $\limsup_{|\omega|\to\infty} |\omega|^{-1/\theta} \|(i\omega - \mathcal{A})^{-1}\| < \infty$.
  • Proof Strategy: By contradiction, they assume the resolvent condition fails. This implies the existence of a sequence of frequencies $\lambda_n \to \infty$ and normalized states $U_n$ such that $(i\lambda_n - \mathcal{A})U_n \to 0$.
  • Through a series of multiplier techniques (multiplying equations by $2xu_x$, $2(x+1)w_x$, etc.) and utilizing weighted Poincaré-type inequalities, they derive estimates showing that the norm of the state $U_n$ must tend to zero, contradicting the normalization $\|U_n\|=1$.

3. Key Contributions

1. Coupled Degenerate System: The paper extends stability analysis from single degenerate wave equations to a coupled system where one string is weakly degenerate and the other can be weakly or strongly degenerate.
2. Decoupling of Degeneracy Effects: A significant theoretical finding is that the stability of the entire coupled system depends only on the degeneracy of the controlled string ($\mu_a$) and is independent of the degeneracy of the uncontrolled string ($\mu_b$).
3. Optimal Decay Rate: The authors derive the precise polynomial decay rate $O(t^{-\frac{2+\mu_a}{2}})$, which is determined solely by the degeneracy parameter $\mu_a$ of the right string.
4. Limitation on Strong Degeneracy: The paper clarifies that if the degeneracy at the connection point ($x=0$) is too strong ($\mu_a \ge 1$), the boundary control at $x=1$ cannot effectively stabilize the left string. This is because the inverse of the coefficient $a^{-1}$ ceases to be integrable, breaking the transmission of control energy across the interface.

4. Main Results

• Theorem 4.1: For the system with $0 \le \mu_a < 1$ and $0 \le \mu_b < 2$, the energy decays polynomially. Specifically, there exists a constant $C > 0$ such that for any initial data $U_0$ in the domain of the operator:
  $$\|T(t)U_0\|_{\mathcal{H}} \le C t^{-\frac{2+\mu_a}{2}} \|U_0\|_{D(\mathcal{A})}$$
• Dependence on Parameters:
  • The decay rate is $t^{-\frac{2+\mu_a}{2}}$.
  • As $\mu_a \to 0$ (non-degenerate), the rate approaches $t^{-1}$ (standard for wave equations with boundary damping).
  • As $\mu_a \to 1$ (approaching strong degeneracy), the rate slows to $t^{-1.5}$.
• Necessity of Weak Degeneracy at Interface: The result holds strictly for $0 \le \mu_a < 1$. If $\mu_a \ge 1$, the control signal cannot propagate through the degeneracy point $x=0$ to the left string, preventing global stability.

5. Significance

• Theoretical Advancement: This work fills a gap in the literature regarding the interaction between coupled degenerate systems and boundary control. Previous works focused on single equations or coupled systems with non-degenerate components.
• Physical Insight: The results provide a mathematical justification for the physical intuition that "strong" degeneracy at an interface acts as a barrier to control propagation. This is crucial for modeling materials with variable density or stiffness that vanish at specific points.
• Methodological Robustness: The successful application of the frequency domain method to this specific coupled, degenerate geometry demonstrates the utility of resolvent estimates in handling complex transmission conditions where traditional energy methods might be difficult to construct.

In summary, the paper establishes that a coupled wave system with one degenerate endpoint can be stabilized by a single boundary control, provided the degeneracy at the connection point is "weak" ($\mu_a < 1$). The stability is polynomial, with a rate explicitly dictated by the degree of degeneracy at the controlled end.