Lyapunov Exponential Stability of the Shallow Water Equations in Trapezoidal Channel

Abstract

This paper solves the problem of the exponential stability in L²-norm of Saint-Venant equations linear hyperbolic system for a non-prismatic and non-rectangular channel. We consider the general case of systems containing not only both arbitrary friction and spatially varying slopes but also spatially varying channel dimensions (width and lateral slope), leading to non-uniform stationary states. An explicit quadratic Lyapunov function is constructed as a weighting function for steady-state small perturbations. We then show that local exponential stability of Saint-Venant equations linear system for a trapezoidal channel can be guaranteed in the L²-norm by an appropriate choice of boundary feedback control. Finally, we give explicitly that control.