Spectral analysis of a wave equation with Kelvin-Voigt damping

A vibrating system with some kind of internal damping represents a distributed or passive control. In this article, a wave equation with clamped boundary conditions and internal Kelvin‐Voigt damping is considered. It is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of finite algebraic multiplicity, and the continuous spectrum that is identical to the essential spectrum is an interval on the left real axis. The asymptotic behavior of eigenvalues is presented. A vibrating system with some kind of internal damping represents a distributed or passive control. In this article, a wave equation with clamped boundary conditions and internal Kelvin‐Voigt damping is considered. It is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of finite algebraic multiplicity, and the continuous spectrum that is identical to the essential spectrum is an interval on the left real axis. The asymptotic behavior of eigenvalues is presented.