Global boundary controllability of the de St. Venant equations between steady states
type: Item
instance of: scholarly article
title: Global boundary controllability of the de St. Venant equations between steady states
zbMATH Open document ID: 1032.93030
DOI: 10.1016/S0294-1449(02)00004-5
author: Martin Gugat
author: Günter Leugering
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2003-04-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78572
review text: 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.
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 76B75
zbMATH DE Number: 1901024
zbMATH Keywords: controllability
zbMATH Keywords: nonlinear hyperbolic system
zbMATH Keywords: St. Venant equation
zbMATH Keywords: characteristic form
zbMATH Keywords: channel flow
zbMATH Keywords: boundary controls
Wikidata QID: Q104129314
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
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OpenAlex ID: W2059933128
description: scientific article; zbMATH DE number 1901024