This paper discusses the boundary feedback stabilization of a reaction–diffusion equation with Robin boundary conditions and in the presence of a time-varying state-delay. The proposed control design strategy is based on a finite-dimensional truncated model obtained via a spectral decomposition. By an adequate selection of the number of modes of the original infinite-dimensional system, we show that the design performed on the finite-dimensional truncated model achieves the exponential stabilization of the original infinite-dimensional system. In the presence of distributed disturbances, we show that the closed-loop system is exponentially input-to-state stable with fading memory.

本文讨论了具有Robin边界条件且存在时变状态时滞的反应扩散方程的边界反馈镇定。所提出的控制设计策略基于通过频谱分解获得的有限维截断模型。经原无限维系统的模式的数量充足的选择，我们证明了设计上的有限维截断模型进行实现原无限维系统的指数镇定。在存在分布扰动的情况下，我们表明闭环系统是具有衰落记忆的指数输入到状态稳定的。