American Institute of Mathematical Sciences

Advanced
June  2013, 3(2): 121-142. doi: 10.3934/mcrf.2013.3.121

Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws

Mapundi K. Banda 1, and  Michael Herty 2,

1. 

Applied Mathematics Division, University of Stellenbosch, Stellenbosch 7600, South Africa
2. 

RWTH Aachen University, IGPM, Templergraben 55, 52056 Aachen, Germany

Received  January 2012 Revised  January 2013 Published  March 2013

Suitable numerical discretizations for boundary control problems of systems of nonlinear hyperbolic partial differential equations are presented. Using a discrete Lyapunov function, exponential decay of the discrete solutions of a system of hyperbolic equations for a family of first--order finite volume schemes is proved. The decay rates are explicitly stated. The theoretical results are accompanied by computational results.
Keywords: Conservation laws, Lyapunov functions., controllability.
Mathematics Subject Classification: Primary: 65Mxx, 49M25, 93D05; Secondary: 65M08, 35L53, 35L6.
Citation: 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
References:
[1]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Networks and Heterogenous Media, 1 (2006), 41.  doi: 10.3934/nhm.2006.1.41.  Google Scholar

[2]

G. Bastin, B. Haut, J.-M. Coron and B. D'andréa-Novel, Lyapunov stability analysis of networks of scalar conservation laws,, Netw. Heterog. Media, 2 (2007), 751.  doi: 10.3934/nhm.2007.2.751.  Google Scholar

[3]

M. Cirinà, Boundary controllability of nonlinear hyperbolic systems,, SIAM J. Control, 7 (1969), 198.   Google Scholar

[4]

J.-M. Coron, Local controllability of a 1-D tank containing a fluid modelled by the shallow water equations,, ESAIM:COCV, 8 (2002), 513.  doi: 10.1051/cocv:2002050.  Google Scholar

[5]

J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (2007).   Google Scholar

[6]

J.-M. Coron, G. Bastin and B. d'Andréa-Novel, Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems,, SIAM J. Control. Optim., 47 (2008), 1460.  doi: 10.1137/070706847.  Google Scholar

[7]

J. M. Coron, B. d'Andréa-Novel and G. Bastin, A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations,, in CD-ROM Proceedings of ECC Karlsruhe, (1999).   Google Scholar

[8]

J.-M. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, IEEE Trans. Automat. Control, 52 (2007), 2.  doi: 10.1109/TAC.2006.887903.  Google Scholar

[9]

J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa-Novel and G. Bastin, Boundary feedback control in networks of open channels,, Automatica J. IFAC, 39 (2003), 1365.  doi: 10.1016/S0005-1098(03)00109-2.  Google Scholar

[10]

J. M. Greenberg and T. T. Li, The effect of boundary damping for the quasilinear wave equation,, J. Differential Equations, 52 (1984), 66.  doi: 10.1016/0022-0396(84)90135-9.  Google Scholar

[11]

M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives,, AMO Advanced Modeling and Optimization, 7 (2005), 9.   Google Scholar

[12]

M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks,, ESAIM: Control, 17 (2011), 28.  doi: 10.1051/cocv/2009035.  Google Scholar

[13]

M. Gugat and G. Leugering, Global boundary controllability of the St. Venant equations between steady states,, Annales de l'Institut Henri Poincaré, 20 (2003), 1.  doi: 10.1016/S0294-1449(02)00004-5.  Google Scholar

[14]

M. Gugat, G. Leugering and E. Schmidt, Global controllability between steady supercritical flows in channel networks,, Mathematical Methods in the Applied Sciences, 27 (2004), 781.  doi: 10.1002/mma.471.  Google Scholar

[15]

G. Leugering and E. J. P. G. Schmidt, On the modelling and stabilization of flows in networks of open canals,, SIAM J. Control Optim., 41 (2002), 164.  doi: 10.1137/S0363012900375664.  Google Scholar

[16]

T. Li, Modelling traffic flow with a time-dependent fundamental diagram,, Math. Meth. Appl. Sci., 27 (2004), 583.  doi: 10.1002/mma.470.  Google Scholar

[17]

T. Li and B. Rao, Exact controllability for first order quasilinear hyperbolic systems with vertical characteristics,, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 980.  doi: 10.1016/S0252-9602(09)60089-8.  Google Scholar

[18]

T. Li, B. Rao and Z. Wang, Contrôlabilité observabilité unilatérales de systèmes hyperboliques quasi-linéaires,, C. R. Math. Acad. Sci. Paris, 346 (2008), 1067.  doi: 10.1016/j.crma.2008.09.004.  Google Scholar

[19]

T. Li and L. Yu, Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues,, Chinese Ann. Math. Ser. B, 24 (2003), 415.  doi: 10.1142/S0252959903000414.  Google Scholar

[20]

T. T. Li, Exact controllability of quasilinear hyperbolic equations (or systems),, Appl. Math. J. Chinese Univ. Ser. A, 20 (2005), 127.   Google Scholar

[21]

M. Slemrod, Boundary feedback stabilization for a quasilinear wave equation,, in, 54 (1983), 221.  doi: 10.1007/BFb0043951.  Google Scholar

[22]

C.-Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems,, ESAIM Control Optim. Calc. Var., 7 (2002), 421.  doi: 10.1051/cocv:2002062.  Google Scholar

[23]

E. Zuazua, Controllability of partial differential equations: Some results and open problems,, in, (2007), 527.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

show all references

References:
[1]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Networks and Heterogenous Media, 1 (2006), 41.  doi: 10.3934/nhm.2006.1.41.  Google Scholar

[2]

G. Bastin, B. Haut, J.-M. Coron and B. D'andréa-Novel, Lyapunov stability analysis of networks of scalar conservation laws,, Netw. Heterog. Media, 2 (2007), 751.  doi: 10.3934/nhm.2007.2.751.  Google Scholar

[3]

M. Cirinà, Boundary controllability of nonlinear hyperbolic systems,, SIAM J. Control, 7 (1969), 198.   Google Scholar

[4]

J.-M. Coron, Local controllability of a 1-D tank containing a fluid modelled by the shallow water equations,, ESAIM:COCV, 8 (2002), 513.  doi: 10.1051/cocv:2002050.  Google Scholar

[5]

J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (2007).   Google Scholar

[6]

J.-M. Coron, G. Bastin and B. d'Andréa-Novel, Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems,, SIAM J. Control. Optim., 47 (2008), 1460.  doi: 10.1137/070706847.  Google Scholar

[7]

J. M. Coron, B. d'Andréa-Novel and G. Bastin, A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations,, in CD-ROM Proceedings of ECC Karlsruhe, (1999).   Google Scholar

[8]

J.-M. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, IEEE Trans. Automat. Control, 52 (2007), 2.  doi: 10.1109/TAC.2006.887903.  Google Scholar

[9]

J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa-Novel and G. Bastin, Boundary feedback control in networks of open channels,, Automatica J. IFAC, 39 (2003), 1365.  doi: 10.1016/S0005-1098(03)00109-2.  Google Scholar

[10]

J. M. Greenberg and T. T. Li, The effect of boundary damping for the quasilinear wave equation,, J. Differential Equations, 52 (1984), 66.  doi: 10.1016/0022-0396(84)90135-9.  Google Scholar

[11]

M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives,, AMO Advanced Modeling and Optimization, 7 (2005), 9.   Google Scholar

[12]

M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks,, ESAIM: Control, 17 (2011), 28.  doi: 10.1051/cocv/2009035.  Google Scholar

[13]

M. Gugat and G. Leugering, Global boundary controllability of the St. Venant equations between steady states,, Annales de l'Institut Henri Poincaré, 20 (2003), 1.  doi: 10.1016/S0294-1449(02)00004-5.  Google Scholar

[14]

M. Gugat, G. Leugering and E. Schmidt, Global controllability between steady supercritical flows in channel networks,, Mathematical Methods in the Applied Sciences, 27 (2004), 781.  doi: 10.1002/mma.471.  Google Scholar

[15]

G. Leugering and E. J. P. G. Schmidt, On the modelling and stabilization of flows in networks of open canals,, SIAM J. Control Optim., 41 (2002), 164.  doi: 10.1137/S0363012900375664.  Google Scholar

[16]

T. Li, Modelling traffic flow with a time-dependent fundamental diagram,, Math. Meth. Appl. Sci., 27 (2004), 583.  doi: 10.1002/mma.470.  Google Scholar

[17]

T. Li and B. Rao, Exact controllability for first order quasilinear hyperbolic systems with vertical characteristics,, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 980.  doi: 10.1016/S0252-9602(09)60089-8.  Google Scholar

[18]

T. Li, B. Rao and Z. Wang, Contrôlabilité observabilité unilatérales de systèmes hyperboliques quasi-linéaires,, C. R. Math. Acad. Sci. Paris, 346 (2008), 1067.  doi: 10.1016/j.crma.2008.09.004.  Google Scholar

[19]

T. Li and L. Yu, Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues,, Chinese Ann. Math. Ser. B, 24 (2003), 415.  doi: 10.1142/S0252959903000414.  Google Scholar

[20]

T. T. Li, Exact controllability of quasilinear hyperbolic equations (or systems),, Appl. Math. J. Chinese Univ. Ser. A, 20 (2005), 127.   Google Scholar

[21]

M. Slemrod, Boundary feedback stabilization for a quasilinear wave equation,, in, 54 (1983), 221.  doi: 10.1007/BFb0043951.  Google Scholar

[22]

C.-Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems,, ESAIM Control Optim. Calc. Var., 7 (2002), 421.  doi: 10.1051/cocv:2002062.  Google Scholar

[23]

E. Zuazua, Controllability of partial differential equations: Some results and open problems,, in, (2007), 527.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

[1]

Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751
[2]

Adimurthi , Shyam Sundar Ghoshal, G. D. Veerappa Gowda. Exact controllability of scalar conservation laws with strict convex flux. Mathematical Control & Related Fields, 2014, 4 (4) : 401-449. doi: 10.3934/mcrf.2014.4.401
[3]

Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[4]

Mauro Garavello. A review of conservation laws on networks. Networks & Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565
[5]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[6]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[7]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
[8]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143
[9]

Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure & Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755
[10]

Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73
[11]

Peter Giesl, Sigurdur Hafstein. Computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : i-ii. doi: 10.3934/dcdsb.2015.20.8i
[12]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41
[13]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
[14]

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
[15]

Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673
[16]

Rinaldo M. Colombo, Kenneth H. Karlsen, Frédéric Lagoutière, Andrea Marson. Special issue on contemporary topics in conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : i-ii. doi: 10.3934/nhm.2016.11.2i
[17]

Laurent Lévi, Julien Jimenez. Coupling of scalar conservation laws in stratified porous media. Conference Publications, 2007, 2007 (Special) : 644-654. doi: 10.3934/proc.2007.2007.644
[18]

Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617
[19]

Alexander Bobylev, Mirela Vinerean, Åsa Windfäll. Discrete velocity models of the Boltzmann equation and conservation laws. Kinetic & Related Models, 2010, 3 (1) : 35-58. doi: 10.3934/krm.2010.3.35
[20]

Gui-Qiang Chen, Monica Torres. On the structure of solutions of nonlinear hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1011-1036. doi: 10.3934/cpaa.2011.10.1011

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences