This article employs the observer-based output feedback control technique to deal with dynamic compensator design for a linear $N$ -D parabolic partial differential equation (PDE) with multiple local piecewise control inputs and multiple noncollocated local piecewise observation outputs. These control inputs and observation outputs are provided by only few actuators and noncollocated sensors active over partial areas (or entire) of the spatial domain. An observer-based dynamic feedback compensator is constructed for exponential stabilization of the linear PDE. Poincaré–Wirtinger inequality and its variant in $N$ -D spatial domain are presented for the closed-loop stability analysis. By the Lyapunov direct method and Poincaré–Wirtinger inequality and its variant, sufficient conditions on the existence of such feedback compensator of the linear PDE are developed, and presented in term of standard linear matrix inequalities. Well-posedness is also analyzed for both open-loop PDE and resulting closed-loop coupled PDEs within the $C_0$ -semigroup framework. Finally, numerical simulation results are presented to support the proposed design method.

中文翻译：

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线性抛物线MIMO PDE的动态补偿器设计。 $ N $维空间域

本文采用基于观测器的输出反馈控制技术来处理线性的动态补偿器设计 $ N $ -D抛物线偏微分方程（PDE），具有多个局部分段控制输入和多个非并置局部分段观测输出。这些控制输入和观察输出仅由在空间域的部分区域（或整个区域）上活动的少数执行器和非并置传感器提供。基于观察者的动态反馈补偿器可用于线性PDE的指数稳定。Poincaré–Wirtinger不等式及其在$ N $ 提出了-D空间域用于闭环稳定性分析。通过Lyapunov直接法和庞加莱-维特林格不等式及其变式，为线性PDE的这种反馈补偿器的存在开发了充分的条件，并用标准线性矩阵不等式表示。还分析了开环PDE以及由此产生的闭环耦合PDE的适定性$ C_0 $ -semigroup框架。最后，给出了数值仿真结果以支持所提出的设计方法。