A Complete Inverse Optimality Study for a Tank-Liquid System

TL;DR

This paper conducts a comprehensive inverse optimality analysis for a linearized tank-liquid system modeled by viscous Saint-Venant equations, establishing existence, uniqueness, and stabilization via feedback laws.

Contribution

It introduces a novel inverse optimality framework for a complex fluid system, including weak solution analysis and the design of optimal stabilizing feedback laws.

Findings

Existence and uniqueness of weak solutions without compatibility conditions

Construction of stabilizing feedback laws using Control Lyapunov Functional

Optimal feedback laws provide stronger stability estimates

Abstract

This paper presents a complete inverse optimality study for a linearized tank-liquid system where the liquid is described by the viscous Saint-Venant model with surface tension and possible wall friction. We define an appropriate weak solution notion for which we establish existence/uniqueness results with inputs that do not necessarily satisfy any compatibility condition as well as stabilization results with feedback laws that are constructed with the help of a Control Lyapunov Functional. We show that the proposed family of stabilizing feedback laws is optimal for a certain meaningful quadratic cost functional. Finally, we show that the optimal feedback law guarantees additional stronger stability estimates which are similar to those obtained in the case of classical solutions.