标题： 适用于一般圣维南方程的PI控制器 显示英文标题

标题： PI controllers for the general Saint-Venant equations

摘要： 我们研究了在$H^{2}$范数下，使用单一比例积分（PI）控制在渠道一端的非线性圣维南（或浅水）方程的指数稳定性，该方程具有任意摩擦和坡度。 通过一个良好但简单的李雅普诺夫函数，我们找到了PI控制增益的一个简单且显式的条件，以确保任何稳态的指数稳定性。 该条件与坡度、摩擦系数、河流长度、流入扰动以及更令人惊讶的是，与所考虑的稳态无关。 当流入扰动是时变的且不存在稳态时，系统仍然具有输入-状态稳定性，并且我们证明稍微改变PI控制可以恢复缓慢变化轨迹的指数稳定性。 显示英文摘要

摘要： We study the exponential stability in the $H^{2}$ norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single Proportional-Integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steady-state considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the Input-to-State stability of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.