基于波方程边界反馈控制系统稳定性的谱分析

The Spectral Analysis of Stability Based on the Wave System with Boundary Feedback Control

对具反馈边界控制问题中的一维带黏性阻力和恢复力的波方程系统,在函数空间中将其化为抽象Cauchy问题,其系统算子具预解紧且是某C0压缩半群的生成元,得到了系统算子的谱分布特征,包括其谱点实部的上确界小于零以及谱点位于某平行于虚轴的有限带域中,为进一步讨论该系统的稳定性、可控性及Riesz基性质奠定了基础.

For one-dimensional wave equation with viscous damping and a restoring force in the problem of boundary feedback control, we abstract it as a Cauchy problem in the functional space. We give that the system operator generates a C0-semigroup and has compact resolvent. Furthermore, we get the spectral feature of the system operator, including the spectrum, which lies in the left half plane, and it is in a limited strip region paralleled to the imaginary axis. The work paves a ground for further discussing the stability, controllability and Riesz base＇s properties of the system.