研究具有Dirichlet边界条件和内部时滞控制的热方程的镇定问题. 文章的目标是设计一个状态反馈控制器, 使得闭环系统以指定的衰减率 $\lambda$ 指数衰减. 与早期的控制器设计方法不同, 文章探索一种新的控制器设计方法------对偏微分方程的参数化控制器设计. 首先, 将带有时滞的控制系统转换成由传输方程和热方程构成的串联系统. 然后, 构建一个具有指数稳定性并且与文章所研究的系统具有类似结构的目标系统. 最后, 选择合适的核函数, 使其构成的有界线性变换可以将闭环系统映到目标系统. 通过选择不同的核函数, 可以得到由目标系统到闭环系统的逆变换.

In this paper, we study the stabilization problem of a heat equation with Dirichlet boundary conditions and internal delayed control. Our goal is to design a state feedback control such that the closed-loop system decays exponentially at designated decay rate $\lambda$. Different from earlier approach of controller design, in the present paper, we explore a new approach of controller design --- The paramerization controller for Partial Differential Equations (PDEs). Firstly, we formulate the system with delayed control into a cascaded system of a transport equation and heat equation. Then, we construct a target system that has designated stability and a similar structure as the system under consideration. Finally, we select the suitable kernel functions of parameterization controller that forms a bounded linear transformation and maps the closed-loop system to the target system. By selecting different kernel function of controller we obtain the inverse transformation from the target system and the closed-loop system.