Article Contents
Article Contents

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

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

 Citation:

•  [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.