A link between Riemann invariants and frequency domain approaches for boundary control of open channel flow

Open channel flow is traditionally described by hyperbolic conservation laws (the Saint-Venant equations), that can be controlled using boundary conditions. For horizontal frictionless channels, a classical approach consists in using the characteristic form to diagonalize the equations, using so-called Riemann invariants. This elegant approach is much more difficult to apply when friction and slope are not zero, i.e. in the vast majority of cases. On the other hand, a Laplace based method enables to diagonalize the system with nonzero slope and friction, but in the frequency domain. This paper enlightens a link between both methods, showing that the frequency domain method can be considered as an extension of the Riemman invariants form for channels with non zero slope and friction. As an application, we derive explicit expressions for the boundary controls solving the motion planning problem.