一类具有时变系数的抛物系统的积分输入状态镇定问题

Integral input-to-state stabilization of a class of parabolic systems with time-varying coefficients

中文关键词  积分输入状态稳定性  反步法  镇定  Lyapunov逼近方法  比较原理  抛物方程  时变系数

英文关键词  integral input-to-state stability  backstepping  stabilization  approximative Lyapunov method  comparison principle  parabolic equation  time-varying coefficient

中文摘要

      对于具有时变系数的抛物系统, 如何通过非时变的核函数进行边界反馈控制设计以确保闭环系统的稳定性, 一直是极具挑战性的问题. 本文考察一类特殊的具有时空变系数的抛物系统的镇定问题. 具体地, 在未对时变系数施加Gevrey条件也未使用事件触发策略的前提下, 本文通过与时间变量无关的核函数设计一个边界反馈控制器以镇定系统; 为了刻画外加扰动为系统稳定性带来的影响, 本文在输入状态稳定性理论框架下研究闭环系统的稳定性, 特别地, 利用 Lyapunov逼近方法以及带有非局部边界条件的抛物系统的比较原理, 建立闭环系统在空间L1范数下关于扰动的积分输入状态稳定性估计式. 通过数值实验, 进一步验证本文设计的控制器的有效性及所提出的方法的可行性.

英文摘要

      For parabolic systems with time-varying coefficients, it remains a challenging problem how to design a boundary feedback control via a time-invariant kernel function for ensuring the stability of the closed-loop system. In this paper, the problem of stabilization of certain class of parabolic systems with space-time-varying coefficients is investigated. Specifically, without applying a Gevrey condition and an event-triggered scheme, a boundary feedback controller is designed by using a time invariant kernel function. Meanwhile, in order to characterize the influence of external disturbances on the stability of the system, the stability of the closed-loop system is studied in the framework of input-to-state stability theory (ISS theory). In particular, the L1-ISS of the considered system is established in the spatial L1-norm by using the approximative Lyapunov method and comparison principle for parabolic PDEs with nonlocal boundary conditions. The validity of the controller and the proposed approach are further verified by numerical simulations.