关于具间接阻尼的强耦合波动方程组的边界耗散传输的敏感性

On the sensibility of the transmission of boundary dissipation for strongly coupled and indirectly damped systems of wave equations

本文考察由两个强耦合的波动方程组成的间接边界阻尼反馈系统的稳定性,阐明此类系统的稳定性依赖于耦合的类型、无阻尼系统是否有隐含正则性、以及边界耗散的阶数和无阻尼边界条件间的匹配等诸多因素.首先证明,当无阻尼边界为Dirichlet边界条件时,系统是一致指数稳定的;而当其为Neumann边界条件时,只能建立系统的多项式稳定性.其次,通过谱分析的方法,揭示间接边界阻尼反馈系统的能量在方程间的传递与无阻尼边界条件之间的内在联系.

We consider the stability of a system of two strongly coupled wave equations by means of only one boundary feedback.We show that the stability of the system depends in a very complex way on all of the involved factors such as the type of coupling,the hinged regularity and the accordance of boundary conditions.We first show that the system is uniformly exponentially stable if the undamped equation has Dirichlet boundary condition,while it is only polynomially stable if the undamped equation is subject to Neumann boundary condition.Next,by a spectral approach,we show that this sensibility of stability with respect to the boundary conditions on the undamped equation is intrinsically linked with the transmission of the vibration as well as the dissipation between the equations.