Control of a Linearized Viscous Liquid–Tank System with Surface Tension

.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.KeywordsSaint-Venant modelshallow water equationsfeedback stabilizationcontrol Lyapunov functionalhigher-order partial differential equationsMSC codes93C2093B5293D3035G4676D5576D4576D27