This paper investigates the guaranteed cost design problem for controlling a class of semilinear parabolic partial differential equation (PDE) systems using mobile collocated actuators and sensors. Initially, a mode indicator function is employed to indicate the different modes for all actuator/sensor pairs according to whether each actuator/sensor pair is static or mobile. Subsequently, a mode-dependent switching control scheme is proposed and the well-posedness of the closed-loop PDE system is also analysed. Then, based on Lyapunov direct method, an integrated design of switching controllers and mobile actuator/sensor guidance laws is developed in the form of linear matrix inequalities, such that the closed-loop PDE system is exponentially stable while providing an upper bound for the prescribed quadratic cost function. Moreover, a suboptimal guaranteed cost design problem is also addressed to make the cost bound as small as possible. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed design method.

本文研究了使用移动并置执行器和传感器控制一类半线性抛物偏微分方程 (PDE) 系统的保证成本设计问题。最初，使用模式指示器功能来根据每个致动器/传感器对是静态的还是移动的来指示所有致动器/传感器对的不同模式。随后，提出了一种与模式相关的切换控制方案，并分析了闭环偏微分方程系统的适定性。然后，基于 Lyapunov 直接法，以线性矩阵不等式的形式开发了开关控制器和移动执行器/传感器引导律的集成设计，使得闭环 PDE 系统在提供规定上限的同时呈指数稳定二次成本函数。而且，还解决了次优保证成本设计问题，以使成本界限尽可能小。最后，数值模拟被提出来说明所提出的设计方法的有效性。