Exponential stability of PI control for Saint-Venant equations with a friction term

We consider open channels represented by Saint-Venant equations that are monitored and controlled at the downstream boundary and subject to unmeasured flow disturbances at the upstream boundary. We address the issue of feedback stabilization and disturbance rejection under Proportional-Integral (PI) boundary control. For channels with uniform steady states, the analysis has been carried out previously in the literature with spectral methods as well as with Lyapunov functions in Riemann coordinates. In this article, our main contribution is to show how the analysis can be extended to channels with non-uniform steady states with a Lyapunov function in physical coordinates. Introduction The hyperbolic Saint-Venant equations are commonly used for the description of water flow dynamics in open channels and for the design of management and control systems in irrigation networks and navigable rivers. In particular, the exponential stabilization of Saint-Venant equations by boundary feedback control has been a recurring research topic in the literature for more than twenty years. The earlier results dealt with static proportional control. In the simplest case of horizontal channels with negligible friction, the stability analysis was carried out in [6] with an entropy Lyapunov function, in [16, 11] with the method of characteristics, and in [7, Section VI] with a Lyapunov function in Riemann coordinates. The stability analysis was then extended to channels with slope and friction. In the special case of a uniform steady state, the stability analysis was carried out with a spectral method for linearized equations in [17, Section 6]. However the linearized system stability does not directly imply the stability of the steady state for the nonlinear SaintVenant equations (see e.g. [8]). For this nonlinear case, the stability analysis is done in [4, 13] with a Lyapunov function in Riemann coordinates. More recently, the case of channels with friction and slope and non-uniform steady state was considered in [3] and [15] with dedicated Lyapunov functions expressed in physical coordinates. The boundary feedback stabilization of Saint-Venant equations by Proportional-Integral (PI) control has received much less attention in the literature. It has been analyzed for channels with uniform steady states in [5] with a spectral method and in [14, Section 4], [2, Section 5.5] with Lyapunov functions in Riemann coordinates. In the present article, our main contribution is to show how the analysis of [3] can be extended to channels with non-uniform steady states under PI control, using a Lyapunov function in physical coordinates. Obviously, in principle, stabilization is also possible with more sophisticated control laws. In particular, the recent backstepping method for 2 × 2 hyperbolic systems, see e.g. [10, 1, 12], ∗Department of Mathematical Engineering, ICTEAM, University of Louvain, Louvain-La-Neuve, Belgium. †Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, LJLL, équipe CAGE, F-75005 Paris, France.