Control of systems of conservation laws with boundary errors

The general problem under consideration in this paper is the stability analysis of hyperbolic systems. Some sufficient criteria on the boundary conditions exist for the stability of a system of conservation laws. We investigate the problem of the stability of such a system in presence of boundary errors that have a small $\mathcal{C}^1$-norm. Two types of perturbations are considered in this work: the errors proportional to the solutions and those proportional to the integral of the solutions. We exhibit a sufficient criterion on the boundary conditions such that the system is locally exponentially stable with a robustness issue with respect to small boundary errors. We apply this general condition to control the dynamic behavior of a pipe filled with water. The control is defined as the position of a valve at one end of the pipe. The potential application is the study of hydropower installations to generate electricity. For this king of application it is important to avoid the waterhammer effect and thus to control the $\mathcal{C}^1$-norm of the solutions. Our damping condition allows us to design a controller so that the system in closed loop is locally exponential stable with a robustness issue with respect to small boundary errors. Since the boundary errors allow us to define the stabilizing controller, small errors in the actuator may be considered. Also a small integral action to avoid possible offset may also be added.