Feedback Stabilization for a Scalar Conservation Law with PID Boundary Control

• Published in: Chinese annals of mathematics. Serie B Vol. 36; no. 5; pp. 763 - 776
• Main Authors: Coron, Jean Michel, Tamasoiu, Simona Oana
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2015; Sorbonne Universités, UPMC University Paris 06, NUMR 7598 Laboratoire Jacques-Louis Lions, F-75005,Paris, France
• Subjects: Applications of Mathematics; Mathematics; Mathematics and Statistics; 反馈镇定; 守恒律; 指数稳定性; 标量; 比例积分微分; 积分控制器; 稳定性分析; 边界控制; PID controllers; 35L50; Boundary feedback; 93D15; Linear scalar conservation law; 93B52
• ISSN: 0252-9599; 1860-6261
• DOI: 10.1007/s11401-015-0975-8

This paper deals with a scalar conservation law in 1-D space dimension, and in particular, the focus is on the stability analysis for such an equation. The problem of feedback stabilization under proportional-integral-derivative （PID for short） boundary control is addressed. In the proportional-integral （PI for short） controller case, by spectral analysis, the authors provide a complete characterization of the set of stabilizing feedback parameters, and determine the corresponding time delay stability interval. Moreover, the stability of the equilibrium is discussed by Lyapunov function techniques, and by this approach the exponential stability when a damping term is added to the classical PI controller scheme is proved. Also, based on Pontryagin results on stability for quasipolynomials, it is shown that the closed-loop system subject to PID control is always unstable.