Motivation & Objective

To address the destabilizing effect of time delay in wave equations, which can destroy well-posedness or stability even for arbitrarily small delays.
To restore well-posedness and stability in the presence of delay by introducing a Kelvin-Voigt damping term $a\,BB^{*}u'$ with $a>0$.
To establish exponential stability for the abstract wave equation with delay under the condition $\tau \leq a$, using a frequency-domain method.
To prove that the condition $\tau \leq a$ is optimal for exponential stability in the given framework.
To provide a general abstract framework applicable to PDEs such as the wave equation with delayed internal damping.

Proposed method

Transform the second-order wave equation with delay into a first-order system using an auxiliary variable $z(\rho,t) = B^{*}u(t - \tau\rho)$ for $\rho \in (0,1)$.
Define a new state vector $U = (u, u', z)^T$ and reformulate the system as an abstract evolution equation $U' = \mathcal{A}U$ in a Hilbert space $\mathcal{H}$.
Use semigroup theory to prove that the operator $\mathcal{A}$ generates a $C_0$-semigroup, ensuring well-posedness of the system.
Apply a frequency-domain approach to analyze the spectrum of $\mathcal{A}$, focusing on the resolvent norm and the behavior of eigenvalues.
Perform a detailed spectral analysis to show that the spectrum of $\mathcal{A}$ is contained in a left half-plane and that the resolvent grows slowly at infinity.
Use contradiction arguments based on sequences of normalized eigenvectors to prove that $\mathcal{A}$ generates an exponentially stable semigroup when $\tau \leq a$.

RQ1Can the destabilizing effect of time delay in a wave equation be counteracted by introducing a Kelvin-Voigt damping term?
RQ2Under what conditions on the delay $\tau$ and damping coefficient $a$ is exponential stability achieved for the abstract wave equation with delay?
RQ3Is the condition $\tau \leq a$ optimal for exponential stability in this class of systems?
RQ4How does the addition of a Kelvin-Voigt damping term restore well-posedness and stability in the presence of delay?
RQ5Can the frequency-domain method be effectively applied to analyze the stability of abstract wave equations with delay and damping?

The system (1.1)–(1.3) is exponentially stable in the Hilbert space $\mathcal{H}$ if $\tau \leq a$, where $a > 0$ is the Kelvin-Voigt damping coefficient.
The energy of the system decays exponentially: $E(t) \leq M e^{-\omega t} E(0)$ for some $M, \omega > 0$, with $\omega$ depending on $a$, $\tau$, and the spectral properties.
The condition $\tau \leq a$ is optimal, as exponential stability fails to hold if $\tau > a$, based on spectral analysis and contradiction arguments.
The proof relies on a frequency-domain approach combined with a precise spectral analysis of the infinitesimal generator $\mathcal{A}$, showing that the spectrum is contained in a left half-plane.
Well-posedness is established via semigroup theory by proving that $\mathcal{A} - a_*^{-1}I$ is maximal dissipative for appropriate $a_*$, ensuring the existence of a $C_0$-semigroup.
The result is applied to the concrete wave equation $u_{tt} - a\Delta u_t - \Delta u(x,t - \tau) = 0$ in a bounded domain $\Omega$, with Dirichlet boundary conditions, confirming exponential stability under $\tau \leq a$.

Hisashi Nishiyama· arXiv (Cornell University)· 2015· 4 citations

Masahiro Ikeda et al.· arXiv (Cornell University)· 2012· 1 citations

Jorge Cortés· arXiv (Cornell University)· 2009· 401 citations

Christophe Prieur et al.· arXiv (Cornell University)· 2017· 114 citations

Jean Auriol et al.· arXiv (Cornell University)· 2017· 81 citations

John M. Davis et al.· ArXiv.org· 2009· 67 citations