# American Institute of Mathematical Sciences

June  2009, 4(2): 177-187. doi: 10.3934/nhm.2009.4.177

## On Lyapunov stability of linearised Saint-Venant equations for a sloping channel

 1 Center for Systems Engineering and Applied Mechanics (CESAME), Department of Mathematical Engineering, Université catholique de Louvain, 4, Avenue G. Lemaître, 1348 Louvain-la-Neuve 2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Institut Universitaire de France, 175, Rue du Chevaleret, 75013 Paris, France 3 Centre de Robotique (CAOR), Ecole nationale supérieure des mines de Paris, 60, Boulevard Saint Michel, 75272 Paris Cedex 06

Received  October 2008 Revised  January 2009 Published  June 2009

We address the issue of the exponential stability (in $L^2$-norm) of the classical solutions of the linearised Saint-Venant equations for a sloping channel. We give an explicit sufficient dissipative condition which guarantees the exponential stability under subcritical flow conditions without additional assumptions on the size of the bottom and friction slopes. The stability analysis relies on the same strict Lyapunov function as in our previous paper [5]. The special case of a single pool is first treated. Then, the analysis is extended to the case of the boundary feedback control of a general channel with a cascade of $n$ pools.
Citation: Georges Bastin, Jean-Michel Coron, Brigitte d'Andréa-Novel. On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Networks & Heterogeneous Media, 2009, 4 (2) : 177-187. doi: 10.3934/nhm.2009.4.177
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