Boundary controllability between sub- and supercritical flow

There are several studies of the boundary controllability of quasi-linear hyperbolic systems where it is assumed that the eigenvalues of the system matrix do not change their signs during the controlled process. In this paper we consider the flow through a frictionless horizontal rectangular channel that is governed by de St. Venant equations and show that the state can be controlled in finite time from a stationary initial state to a given stationary terminal state in such a way that during this transition, the state stays in the class of C-1 functions, so in particular no shocks occur. There is no restriction on the initial and terminal state, so in some cases it is necessary that one or both eigenvalues of the system matrix change the sign during the process. Various different cases occur: control between subcritical states, control between supercritical states, transition from a subcritical to a supercritical state, and transition from a supercritical to a subcritical state. In the last two cases of a control between states of a different type, one eigenvalue of the system matrix changes its sign during the process. When this happens at a boundary point during the process, it is necessary to switch the type of boundary conditions. We show how to construct controls where at each boundary at most one such switching is necessary.