Spectral analysis of stability of control system based on wave equation boundary feedback

spectral distribution;  stability;  wave equation;  pre-tightening;  C0 semigroup

For the wave equation system with one-dimensional viscous resistance and restoring force in the feedback boundary control problem, it is transformed into an abstract Cauchy problem in the function space, and the system operator is pre-compacted and is the generator of a certain C0 compression semigroup, and the spectral distribution characteristics of the system operator are obtained, including that the supremum of the real part of the spectral point is less than zero and the spectral point is located in a finite band domain parallel to the imaginary axis, which lays a foundation for further discussion of the stability, controllability and Riesz basis properties of the system.