We first study stabilization of heat equation with globally Lipschitz nonlinearity. We consider the point measurements with constant delay and use spatial decomposition. Inspired by recent developments in the area of ordinary differential equations (ODEs) with time-delays, for the stability analysis, we suggest an augmented Lyapunov functional depending on the state derivative that is based on Legendre polynomials. Global exponential stability conditions are derived in terms of linear matrix inequalities (LMIs) that depend on the degree $\mathcal{N}$ of Legendre polynomials. The stability conditions form a hierarchy of LMIs: if the LMIs hold for $\mathcal{N}$, they hold for $\mathcal{N}+1$. The dual observer design problem with constant delay is also formulated. We further consider stabilization of Korteweg–de Vries–Burgers (KdVB) equation using the point measurements with constant delay. Due to the third-order partial derivative in KdVB equation, the Lyapunov functionals that depend on the state derivative are not applicable here, which is different from the case of heat equation. We suggest a novel augmented Lyapunov functional depending on the state only that leads to improved regional stability conditions in terms of LMIs. Finally, numerical examples illustrate the efficiency of the method.

我们首先研究了具有全局 Lipschitz 非线性的热方程的稳定性。我们考虑具有恒定延迟的点测量并使用空间分解。受时滞常微分方程 (ODE) 领域的最新发展启发，对于稳定性分析，我们建议使用基于勒让德多项式的状态导数的增强李雅普诺夫函数。全局指数稳定性条件是根据线性矩阵不等式 (LMI) 导出的，该不等式取决于度数$\mathcal{N}$勒让德多项式。稳定性条件形成 LMI 的层次结构：如果 LMI 保持$\mathcal{N}$，他们坚持 $\mathcal{N}+1$. 还制定了具有恒定延迟的双观测器设计问题。我们进一步考虑使用具有恒定延迟的点测量来稳定 Korteweg-de Vries-Burgers (KdVB) 方程。由于 KdVB 方程中的三阶偏导数，依赖于状态导数的李雅普诺夫泛函不适用于这里，这与热方程的情况不同。我们建议一种新的增强 Lyapunov 泛函仅取决于状态，从而改善 LMI 方面的区域稳定性条件。最后，数值例子说明了该方法的有效性。