Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control

In this paper, the negative proportional dynamic feedback is designed in the right boundary of the wave component of the 1-d heat-wave system coupled at the interface and the long-time behavior of the system is discussed. The system is formulated into an abstract Cauchy problem on the energy space. The energy of the system does not increase because the semigroup generated by the system operator is contracted. In the meanwhile, the asymptotic stability of the system is derived in light of the spectral configuration of the system operator. Furthermore, the spectral expansions of the system operator are precisely investigated and the asymptotical stability is not exponential and is shown in view of the spectral expansions.