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 and 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), 41-56. doi: 10.3934/nhm.2006.1.41.

[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), 751-759. doi: 10.3934/nhm.2007.2.751.

[3]

M. Cirinà, Boundary controllability of nonlinear hyperbolic systems, SIAM J. Control, 7 (1969), 198-212.

[4]

J.-M. Coron, Local controllability of a 1-D tank containing a fluid modelled by the shallow water equations, ESAIM:COCV, 8 (2002), 513-554. doi: 10.1051/cocv:2002050.

[5]

J.-M. Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.

[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), 1460-1498. doi: 10.1137/070706847.

[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).

[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), 2-11. doi: 10.1109/TAC.2006.887903.

[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), 1365-1376. doi: 10.1016/S0005-1098(03)00109-2.

[10]

J. M. Greenberg and T. T. Li, The effect of boundary damping for the quasilinear wave equation, J. Differential Equations, 52 (1984), 66-75. doi: 10.1016/0022-0396(84)90135-9.

[11]

M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives, AMO Advanced Modeling and Optimization, 7 (2005), 9-37.

[12]

M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 28-51. doi: 10.1051/cocv/2009035.

[13]

M. Gugat and G. Leugering, Global boundary controllability of the St. Venant equations between steady states, Annales de l'Institut Henri Poincaré, Nonlinear Analysis, 20 (2003), 1-11. doi: 10.1016/S0294-1449(02)00004-5.

[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), 781-802. doi: 10.1002/mma.471.

[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), 164-180. doi: 10.1137/S0363012900375664.

[16]

T. Li, Modelling traffic flow with a time-dependent fundamental diagram, Math. Meth. Appl. Sci., 27 (2004), 583-601. doi: 10.1002/mma.470.

[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), 980-990. doi: 10.1016/S0252-9602(09)60089-8.

[18]

T. Li, B. Rao and Z. Wang, Contrôlabilité observabilité unilatérales de systèmes hyperboliques quasi-linéaires, C. R. Math. Acad. Sci. Paris, 346 (2008), 1067-1072. doi: 10.1016/j.crma.2008.09.004.

[19]

T. Li and L. Yu, Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues, Chinese Ann. Math. Ser. B, 24 (2003), 415-422. doi: 10.1142/S0252959903000414.

[20]

T. T. Li, Exact controllability of quasilinear hyperbolic equations (or systems), Appl. Math. J. Chinese Univ. Ser. A, 20 (2005), 127-146.

[21]

M. Slemrod, Boundary feedback stabilization for a quasilinear wave equation, in "Control Theory for Distributed Parameter Systems and Applications" (Vorau, 1982), Lecture Notes in Control and Information Sciences, 54, Springer, Berlin, (1983), 221-237. doi: 10.1007/BFb0043951.

[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), 421-442 (electronic). doi: 10.1051/cocv:2002062.

[23]

E. Zuazua, Controllability of partial differential equations: Some results and open problems, in "Handbook of Differential Equations: Evolutionary Equations. Vol. III" (eds. C. Dafermos and E. Feireisl), Elsevier/North-Holland, Amsterdam, (2007), 527-621. doi: 10.1016/S1874-5717(07)80010-7.

show all references

References:
[1]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Networks and Heterogenous Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41.

[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), 751-759. doi: 10.3934/nhm.2007.2.751.

[3]

M. Cirinà, Boundary controllability of nonlinear hyperbolic systems, SIAM J. Control, 7 (1969), 198-212.

[4]

J.-M. Coron, Local controllability of a 1-D tank containing a fluid modelled by the shallow water equations, ESAIM:COCV, 8 (2002), 513-554. doi: 10.1051/cocv:2002050.

[5]

J.-M. Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.

[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), 1460-1498. doi: 10.1137/070706847.

[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).

[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), 2-11. doi: 10.1109/TAC.2006.887903.

[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), 1365-1376. doi: 10.1016/S0005-1098(03)00109-2.

[10]

J. M. Greenberg and T. T. Li, The effect of boundary damping for the quasilinear wave equation, J. Differential Equations, 52 (1984), 66-75. doi: 10.1016/0022-0396(84)90135-9.

[11]

M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives, AMO Advanced Modeling and Optimization, 7 (2005), 9-37.

[12]

M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 28-51. doi: 10.1051/cocv/2009035.

[13]

M. Gugat and G. Leugering, Global boundary controllability of the St. Venant equations between steady states, Annales de l'Institut Henri Poincaré, Nonlinear Analysis, 20 (2003), 1-11. doi: 10.1016/S0294-1449(02)00004-5.

[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), 781-802. doi: 10.1002/mma.471.

[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), 164-180. doi: 10.1137/S0363012900375664.

[16]

T. Li, Modelling traffic flow with a time-dependent fundamental diagram, Math. Meth. Appl. Sci., 27 (2004), 583-601. doi: 10.1002/mma.470.

[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), 980-990. doi: 10.1016/S0252-9602(09)60089-8.

[18]

T. Li, B. Rao and Z. Wang, Contrôlabilité observabilité unilatérales de systèmes hyperboliques quasi-linéaires, C. R. Math. Acad. Sci. Paris, 346 (2008), 1067-1072. doi: 10.1016/j.crma.2008.09.004.

[19]

T. Li and L. Yu, Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues, Chinese Ann. Math. Ser. B, 24 (2003), 415-422. doi: 10.1142/S0252959903000414.

[20]

T. T. Li, Exact controllability of quasilinear hyperbolic equations (or systems), Appl. Math. J. Chinese Univ. Ser. A, 20 (2005), 127-146.

[21]

M. Slemrod, Boundary feedback stabilization for a quasilinear wave equation, in "Control Theory for Distributed Parameter Systems and Applications" (Vorau, 1982), Lecture Notes in Control and Information Sciences, 54, Springer, Berlin, (1983), 221-237. doi: 10.1007/BFb0043951.

[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), 421-442 (electronic). doi: 10.1051/cocv:2002062.

[23]

E. Zuazua, Controllability of partial differential equations: Some results and open problems, in "Handbook of Differential Equations: Evolutionary Equations. Vol. III" (eds. C. Dafermos and E. Feireisl), Elsevier/North-Holland, Amsterdam, (2007), 527-621. doi: 10.1016/S1874-5717(07)80010-7.

[1]

Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks and 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 and Related Fields, 2014, 4 (4) : 401-449. doi: 10.3934/mcrf.2014.4.401
[3]

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

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

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
[6]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[7]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Heterogeneous Media, 2016, 11 (2) : i-ii. doi: 10.3934/nhm.2016.11.2i
[17]

Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617
[18]

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
[19]

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

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

2021 Impact Factor: 1.141

Tools

Metrics

  • PDF downloads (101)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy