  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.
