This paper discusses dynamic feedback compensator design for a linear parabolic partial differential equation (PDE) with multiple inputs and multiple outputs. Actuating control inputs are provided by actuators distributed over partial areas (or active at specified positions) of the spatial domain, and observation outputs are taken from the non-collocated sensors distributed over partial areas of the spatial domain. An observer-based dynamic feedback compensator is constructed via the observer-based feedback control technique to exponentially stabilize the multi-input–multi-output PDE in the spatial |$\mathscr{L}^2$| norm. By constructing an appropriate Lyapunov function candidate and using two variants of Poincaré–Wirtinger inequality, sufficient conditions on the existence of such observer-based dynamic feedback compensator are developed and presented in terms of linear matrix inequalities. The well posedness of the closed-loop coupled PDEs is also analyzed within the framework of |$C_0$| semigroup theory. Finally, numerical simulation results are given to show the effectiveness of the proposed method.

中文翻译：

---

基于Lyapunov的线性抛物面MIMO PDE动态反馈补偿器设计

本文讨论了具有多个输入和多个输出的线性抛物型偏微分方程（PDE）的动态反馈补偿器设计。致动控制输入由分布在空间域的部分区域（或在指定位置的活动区域）上的致动器提供，观察输出来自分布在空间域的部分区域的非并置传感器。通过基于观察者的反馈控制技术构造基于观察者的动态反馈补偿器，以使空间| $ \ mathscr {L} ^ 2 $ |中的多输入多输出PDE指数稳定。规范。通过构造适当的Lyapunov函数候选者并使用Poincaré-Wirtinger不等式的两个变体，就存在基于此类基于观察者的动态反馈补偿器的充分条件，并根据线性矩阵不等式给出了条件。在| $ C_0 $ |的框架内还分析了闭环耦合PDE的适定性。半群论。最后，数值仿真结果表明了该方法的有效性。