Open‐Loop Control of Shallow Water Waves in a Tube With Moving Boundary in Material‐Fixed Coordinates

Open‐loop control design for a one‐dimensional shallow water model with moving boundary and arbitrary cross section is considered. Despite the moving boundary, the transition to material‐fixed coordinates allows to reformulate the model on a constant spatial domain. The focus is on achieving smooth transitions between steady states. Apart from the transition to material‐fixed coordinates, the control design closely resembles the procedure already developed for the original model. Introducing the water level at the nonactuated boundary as a flat output, the given boundary value problem is linked to a Cauchy boundary value problem. The latter can be solved by the method of characteristics. The parameterization of the solutions of the shallow water equations obtained this way simplifies the control design. Numerical validation of the design relies on energy‐preserving spatial discretization.