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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), 41-56. 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), 751-759. doi: 10.3934/nhm.2007.2.751.  Google Scholar

[3]

M. Cirinà, Boundary controllability of nonlinear hyperbolic systems, SIAM J. Control, 7 (1969), 198-212.  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), 513-554. doi: 10.1051/cocv:2002050.  Google Scholar

[5]

J.-M. Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 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), 1460-1498. doi: 10.1137/070706847.  Google Scholar

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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), 2-11. 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), 1365-1376. 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), 66-75. 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), 9-37.  Google Scholar

[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.  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, Nonlinear Analysis, 20 (2003), 1-11. doi: 10.1016/S0294-1449(02)00004-5.  Google Scholar

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[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.  Google Scholar

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

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[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.  Google Scholar

[20]

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

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

[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.  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), 41-56. 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), 751-759. doi: 10.3934/nhm.2007.2.751.  Google Scholar

[3]

M. Cirinà, Boundary controllability of nonlinear hyperbolic systems, SIAM J. Control, 7 (1969), 198-212.  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), 513-554. doi: 10.1051/cocv:2002050.  Google Scholar

[5]

J.-M. Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 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), 1460-1498. 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), 2-11. 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), 1365-1376. 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), 66-75. 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), 9-37.  Google Scholar

[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.  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, Nonlinear Analysis, 20 (2003), 1-11. 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), 781-802. 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), 164-180. doi: 10.1137/S0363012900375664.  Google Scholar

[16]

T. Li, Modelling traffic flow with a time-dependent fundamental diagram, Math. Meth. Appl. Sci., 27 (2004), 583-601. 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), 980-990. 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), 1067-1072. 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), 415-422. 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), 127-146.  Google Scholar

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

[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.  Google Scholar

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