The goal of this paper is to shed light on new opportunities for extremum seeking through extensions to locally quadratic nonlinear maps with actuator dynamics modeled by cascades of partial differential equations (PDEs). First, we deal with PDEs with input delays such as, for example, the notoriously difficult problem of a wave PDE with input delay where, if the delay is left uncompensated, an arbitrarily short delay destroys the closed-loop stability. Then, we move forward to an even more challenging class of problems for parabolic–hyperbolic cascades of PDEs, coping with a heat equation at the input of a wave PDE. The treatment of such systems with PDE–PDE cascades is performed by means of boundary control. The proposed approach yields (small-gain) loops that make the control design constructive and enables stability analysis with quantitative estimates. Local exponential stability and convergence to a small neighborhood of the unknown extremum point are rigorously guaranteed. This result is achieved by using backstepping transformation and averaging in infinite dimensions. Although we keep the presentation using the classical Gradient extremum seeking, the generalization of the results for the Newton-based approach is also indicated. A numerical example is given to illustrate the effectiveness of the proposed extremum seeking boundary control for compensation of PDE–PDE cascades.

本文的目标是通过对局部二次非线性映射的扩展以及由偏微分方程 (PDE) 级联建模的致动器动力学来揭示寻找极值的新机会。首先，我们处理具有输入延迟的 PDE，例如众所周知的具有输入延迟的波 PDE 的难题，如果延迟未得到补偿，任意短的延迟会破坏闭环稳定性。然后，我们继续处理 PDE 的抛物线-双曲线级联的更具挑战性的一类问题，处理波 PDE 输入处的热方程。用 PDE-PDE 级联处理此类系统是通过边界控制进行的。提议的方法产生（小增益）循环，使控制设计具有建设性，并通过定量估计实现稳定性分析。严格保证局部指数稳定性和收敛到未知极值点的小邻域。这个结果是通过使用反步变换和无限维平均来实现的。尽管我们使用经典的梯度极值寻找来保持表示，但也指出了基于牛顿方法的结果的概括。给出了一个数值例子来说明所提出的极值寻求边界控制对 PDE-PDE 级联补偿的有效性。这个结果是通过使用反步变换和无限维平均来实现的。尽管我们使用经典的梯度极值寻找来保持表示，但也指出了基于牛顿方法的结果的概括。给出了一个数值例子来说明所提出的极值寻求边界控制对 PDE-PDE 级联补偿的有效性。这个结果是通过使用反步变换和无限维平均来实现的。尽管我们使用经典的梯度极值寻找来保持表示，但也指出了基于牛顿方法的结果的概括。给出了一个数值例子来说明所提出的极值寻求边界控制对 PDE-PDE 级联补偿的有效性。