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

Mapundi K. Banda; Michael Herty

doi:10.3934/mcrf.2013.3.121

Article Contents

2013, Volume 3, Issue 2: 121-142. Doi: 10.3934/mcrf.2013.3.121

This issue Previous Article Next Article Stability of the determination of a time-dependent coefficient in parabolic equations

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
2.
RWTH Aachen University, IGPM, Templergraben 55, 52056 Aachen

Received: December 31, 2011

Revised: December 31, 2012

Published: May 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 65Mxx, 49M25, 93D05; Secondary: 65M08, 35L53, 35L65.

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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.