Premium
The spectral analysis and exponential stability of a bi‐directional coupled wave‐ODE system
Author(s) -
Zhao Dongxia,
Wang Junmin,
Lu Lu
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5549
Subject(s) - ode , mathematics , eigenfunction , mathematical analysis , eigenvalues and eigenvectors , essential spectrum , exponential stability , spectrum (functional analysis) , wave equation , exponential function , continuous spectrum , exponential growth , physics , quantum mechanics , nonlinear system
In this paper, an unstable linear time invariant (LTI) ODE system X ̇ ( t ) = A X ( t ) is stabilized exponentially by the PDE compensato—a wave equation with Kelvin‐Voigt (K‐V) damping. Direct feedback connections between the ODE system and wave equation are established: The velocity of the wave equation enters the ODE through the variable v t (1, t ); meanwhile, the output of the ODE is fluxed into the wave equation. It is found that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The continuous spectrum consists of an isolated point − 1 d , and there are two branches of asymptotic eigenvalues: the first branch approaches to − 1 d , and the other branch tends to − ∞ . It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. As a consequence, the spectrum‐determined growth condition and exponential stability of the system are concluded.