A Complete Inverse Optimality Study for a Tank-Liquid System
Iasson Karafyllis, Filippos Vokos, Miroslav Krstic
TL;DR
This work tackles inverse optimality for a linearized tank–liquid system described by the viscous Saint-Venant model with surface tension and wall friction. It develops a weak-solution framework that accommodates discontinuous boundary inputs and constructs a Control Lyapunov Functional to derive stabilizing, $LgV$-type feedback laws. The authors prove that the stabilizing feedback is inverse-optimal for a meaningful quadratic cost $J$, and they establish exponential stabilization of a joint state–control norm with an explicit gain-margin structure, plus stronger stability estimates in a higher-norm akin to ISS for parabolic PDEs. The methodology, relying on energy-based arguments and Galerkin approximations, provides a benchmark approach that can extend to other PDEs with boundary control and motivates future explorations of inverse optimality in PDE settings.
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.