热-板传输系统的稳定性分析

Stability analysis of an interaction system coupling plate equation and heat equation

本文研究由处于相邻区域的板方程和热方程构成的耦合系统的稳定性质,其中耦合来自两个区域的交界面上的传输边界条件.在该传输系统中,热方程起着控制器的作用,且耗散通过交界面传输并影响板方程.文献[1]证明了在板方程上施加额外的控制器时,该二维传输系统的能量呈指数衰减.通过应用频域方法,椭圆方程的正则性理论等,可以得到:仅由热方程的耗散即可使得闭环系统指数稳定.这一指数稳定的结论与相应的一维传输系统的性质吻合.最后,文章还分析了不同传输边界条件下的板-热耦合系统的稳定性.

In this paper,we study the stability of a coupled system which consists of a plate equation and a heat equation in adjacent domains,where the coupling comes from the transmission boundary conditions at the interface of the two domains.The heat equation is considered as the controller of the whole system.The dissipative damping produced in the heat equation affects the plate equation via the boundary connections.It is known that the energy of this two-dimensional system decays exponentially when there is an additional dissipation in the boundary of the plate part[1].In this paper,we obtain the exponential stability of the system with dissipation only in heat equation with the help of the frequency domain method,theory of elliptic equations,etc.It is consistent with the stability properties of corresponding one-dimensional beam-heat interaction system.Finally,we also analyze several plate-heat interaction systems with different coupling conditions.