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

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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

Abstract
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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.
[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.
[3]	M. Cirinà, Boundary controllability of nonlinear hyperbolic systems,, SIAM J. Control, 7 (1969), 198.
[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.
[5]	J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (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. 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. 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. 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. 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.
[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.
[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.
[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.
[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.
[16]	T. Li, Modelling traffic flow with a time-dependent fundamental diagram,, Math. Meth. Appl. Sci., 27 (2004), 583. 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. 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. 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. 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.
[21]	M. Slemrod, Boundary feedback stabilization for a quasilinear wave equation,, in, 54 (1983), 221. 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. doi: 10.1051/cocv:2002062.
[23]	E. Zuazua, Controllability of partial differential equations: Some results and open problems,, in, (2007), 527. 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. 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. doi: 10.3934/nhm.2007.2.751.
[3]	M. Cirinà, Boundary controllability of nonlinear hyperbolic systems,, SIAM J. Control, 7 (1969), 198.
[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.
[5]	J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (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. 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. 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. 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. 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.
[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.
[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.
[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.
[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.
[16]	T. Li, Modelling traffic flow with a time-dependent fundamental diagram,, Math. Meth. Appl. Sci., 27 (2004), 583. 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. 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. 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. 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.
[21]	M. Slemrod, Boundary feedback stabilization for a quasilinear wave equation,, in, 54 (1983), 221. 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. doi: 10.1051/cocv:2002062.
[23]	E. Zuazua, Controllability of partial differential equations: Some results and open problems,, in, (2007), 527. doi: 10.1016/S1874-5717(07)80010-7.

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