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

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.
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.

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