Global boundary controllability of the de St. Venant equations between steady states
The problem of exactly controlling the states of the St. Venant equations from a given constant state to another constant state by applying nonlinear boundary controls is considered. The St. Venant equations governing the flow of water in that channel are given in the form of the following quasilinear hyperbolic system: \[ \partial_t\begin{pmatrix} a\\ U\end{pmatrix}+ \begin{pmatrix} U & a\\ g/b(a) & U\end{pmatrix} \partial_x\begin{pmatrix} a\\ U\end{pmatrix}= 0,\tag{1} \] where \(U(t,x)\) and \(a(t,x)\) denote the average velocity over the cross section of the channel and the wetted area at \(x\in [0, L]\) and time \(t\), respectively. Main result: A result on global controllability of (1) with restrictions on the distance between the initial state and the target is presented. The authors show that starting from a constant subcritical initial state we can reach any other constant subcritical state in finite time with boundary controls that can be chosen in such way that shocks are avoided and the state remains within the context of classical solutions. All proofs are based on the characteristic form of the equations. They are constructive and show how controls can actually be computed numerically. As the boundary controls are not unique, there is some flexibility with respect to optimization. The authors' result is global in the sense that it holds for initial states and terminal states that can be arbitrarily far away from each other. This is in contrast to the local results given in the literature that hold for initial states in a neighborhood of the desired terminal state only.
scientific article; zbMATH DE number 1901024