SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1034-1059, April 2024.
Abstract. This paper studies the linearization of the viscous tank–liquid system. The linearization of the tank–liquid system gives a high-order partial differential equation, which is a combination of a wave equation with Kelvin–Voigt damping and a Euler–Bernoulli beam equation. The single input appears in two of the boundary conditions (boundary input). The paper provides results both for the open-loop system (existence/uniqueness of solutions and stability properties of the open-loop system) as well as results for the construction of feedback stabilizers. More specifically, the feedback design methodology is based on control Lyapunov functionals (CLFs). The proposed CLFs are modifications and augmentations of the total energy functionals for the tank–liquid system so that the dissipative effects of viscosity, friction, and surface tension are captured. By focusing on the linearized water–tank system, we are able to provide results that are not provided in the nonlinear case: (1) existence and uniqueness of solutions, (2) simultaneous presence of friction and surface tension, and (3) stabilization in a stronger norm, using a different CLF.

中文翻译：

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具有表面张力的线性粘性液体槽系统的控制

SIAM 控制与优化杂志，第 62 卷，第 2 期，第 1034-1059 页，2024 年 4 月。
摘要。本文研究了粘性罐-液体系统的线性化。罐-液系统的线性化给出了一个高阶偏微分方程，它是带有 Kelvin-Voigt 阻尼的波动方程和 Euler-Bernoulli 梁方程的组合。单个输入出现在两个边界条件（边界输入）中。本文提供了开环系统的结果（解的存在/唯一性和开环系统的稳定性属性）以及反馈稳定器的构造结果。更具体地说，反馈设计方法基于控制李雅普诺夫泛函 (CLF)。所提出的 CLF 是对罐液系统总能量泛函的修改和增强，以便捕获粘度、摩擦和表面张力的耗散效应。通过关注线性化水箱系统，我们能够提供非线性情况下无法提供的结果：（1）解的存在性和唯一性，（2）摩擦力和表面张力同时存在，以及（3）稳定性在更强的规范中，使用不同的 CLF。