带有时滞的非线性Korteweg-de Vries-Burgers方程 和Kuramoto-Sivashinsky方程的输出反馈镇定

In recent years, distributed stabilisation of nonlinear partial differential equations (PDEs) system governed by Korteweg-de Vries-Burgers (KdVB) equation and Kuramoto-Sivashinsky (KS) equation with time delay has attracted a lot of attention, owing to their potential applications in physics. This project is devoted to dealing with two problems: distributed stabilisation of KdVB equation in the presence of constant input delay, and distributed sampled-data control of KS equation with output delay. The delay may be uncertain, but bounded by a known upper bound. The objective of the project is to design a regionally stabilizing controller applied through distributed in space shape functions, on the basis of spatially distributed (either point or averaged) measurements. The existing Lyapunov-Krasovskii functionals for heat equation that depend on the state derivative are not applicable to KdVB equation or KS equation because of the high order spatial derivative and the nonlinearity. The situation becomes more complicated. We will try to solve this problem by suggesting an appropriate Lyapunov-Krasovskii functional that leads to regional stability conditions of the closed-loop system in terms of linear matrix inequalities (LMIs). This could be an important step toward the development of the theory for distributed parameter system and the application in electrical engineering.

近年来，由于潜在的物理应用前景，以带有时滞的Korteweg-de Vries-Burgers（KdVB）方程和Kuramoto-Sivashinsky（KS）方程所描述的非线性偏微分方程系统的分布式镇定控制问题引起了广泛关注。本项目拟解决两个问题：带有输入常时滞的KdVB方程的分布式镇定问题，以及带有输出时滞的KS方程的采样控制问题，其中时滞是不确定的但有上界。该项目主要研究通过点量测输出或均值量测输出，设计使得系统区域稳定的输出反馈控制器。由于KdVB方程和KS方程含有高阶导数项与非线性项，对热方程构造的Lyapunov-Krasovskii函数并不适用于KdVB方程和KS方程系统的稳定性分析中。拟解决的途径是构造适当的Lyapunov-Krasovskii函数并利用LMI方法来推导使得闭环系统区域稳定的条件。该项目无论对分布参数系统理论本身的发展，还是对工程实际应用都具有重要意义。

近年来，由于潜在的物理应用前景，以Burger方程、Korteweg-de Vries（KdV）方程、Korteweg-de Vries-Burgers（KdVB）方程和Kuramoto-Sivashinsky（KS）方程所描述的非线性偏微分方程系统的分布式镇定控制问题引起了广泛关注。本项目主要研究内容包括：1）带有输入常时滞的KdVB方程的分布式镇定问题 2）KS方程的采样控制问题 3）KdV的事件触发控制问题 4）Burgers方程的约束控制问题等。该项目主要研究通过点量测输出或均值量测输出，设计使得系统区域稳定的输出反馈控制器。由于KdV方程、Burger方程、KdVB方程、KS方程含有高阶导数项与非线性项，这给系统的适定性分析与稳定性分析带来巨大的挑战。解决的途径是构造适当的Lyapunov-Krasovskii函数来推导使得闭环系统稳定的条件。该项目无论对分布参数系统理论本身的发展，还是对工程实际应用都具有重要意义。