American Institute of Mathematical Sciences

Advanced
June  2009, 4(2): 393-407. doi: 10.3934/nhm.2009.4.393

Control of systems of conservation laws with boundary errors

Christophe Prieur 1,

1. 

LAAS-CNRS, University of Toulouse, 7, avenue du Colonel Roche, 31077 Toulouse, France

Received  September 2008 Revised  February 2009 Published  June 2009

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.
Keywords: Hyperbolic system, Boundary control, Robust stabilization., Conservation law.
Mathematics Subject Classification: Primary: 35L65, 34D05; Secondary: 35L50, 93D2.
Citation: Christophe Prieur. Control of systems of conservation laws with boundary errors. Networks & Heterogeneous Media, 2009, 4 (2) : 393-407. doi: 10.3934/nhm.2009.4.393
[1]

Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717
[2]

Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control & Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121
[3]

Jean-Michel Coron, Matthias Kawski, Zhiqiang Wang. Analysis of a conservation law modeling a highly re-entrant manufacturing system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1337-1359. doi: 10.3934/dcdsb.2010.14.1337
[4]

Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004
[5]

Afaf Bouharguane. On the instability of a nonlocal conservation law. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 419-426. doi: 10.3934/dcdss.2012.5.419
[6]

Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control & Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761
[7]

A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289
[8]

Zuowei Cai, Jianhua Huang, Liu Yang, Lihong Huang. Periodicity and stabilization control of the delayed Filippov system with perturbation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019235
[9]

Yanning Li, Edward Canepa, Christian Claudel. Efficient robust control of first order scalar conservation laws using semi-analytical solutions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 525-542. doi: 10.3934/dcdss.2014.7.525
[10]

Mohammad Akil, Ali Wehbe. Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions. Mathematical Control & Related Fields, 2019, 9 (1) : 97-116. doi: 10.3934/mcrf.2019005
[11]

Stefano Bianchini. On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 329-350. doi: 10.3934/dcds.2000.6.329
[12]

Gilbert Peralta, Karl Kunisch. Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3055-3083. doi: 10.3934/dcds.2018133
[13]

Alberto Bressan, Graziano Guerra. Shift-differentiabilitiy of the flow generated by a conservation law. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 35-58. doi: 10.3934/dcds.1997.3.35
[14]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255
[15]

Robert I. McLachlan, G. R. W. Quispel. Discrete gradient methods have an energy conservation law. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1099-1104. doi: 10.3934/dcds.2014.34.1099
[16]

Julien Jimenez. Scalar conservation law with discontinuous flux in a bounded domain. Conference Publications, 2007, 2007 (Special) : 520-530. doi: 10.3934/proc.2007.2007.520
[17]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763
[18]

Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050
[19]

Shaohong Fang, Jing Huang, Jinying Ma. Stabilization of a discrete-time system via nonlinear impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020106
[20]

Hongyun Peng, Zhi-An Wang, Kun Zhao, Changjiang Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic & Related Models, 2018, 11 (5) : 1085-1123. doi: 10.3934/krm.2018042

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences