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December  2010, 5(4): 691-709. doi: 10.3934/nhm.2010.5.691

## Classical solutions and feedback stabilization for the gas flow in a sequence of pipes

 1 Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen, Germany, Germany

Received  February 2010 Revised  August 2010 Published  November 2010

We consider the subcritical flow in gas networks consisting of a finite linear sequence of pipes coupled by compressor stations. Such networks are important for the transportation of natural gas over large distances to ensure sustained gas supply. We analyse the system dynamics in terms of Riemann invariants and study stationary solutions as well as classical non-stationary solutions for a given finite time interval. Furthermore, we construct feedback laws to stabilize the system locally around a given stationary state. To do so we use a Lyapunov function and prove exponential decay with respect to the $L^2$-norm.
Citation: Markus Dick, Martin Gugat, Günter Leugering. Classical solutions and feedback stabilization for the gas flow in a sequence of pipes. Networks & Heterogeneous Media, 2010, 5 (4) : 691-709. doi: 10.3934/nhm.2010.5.691
##### References:
 [1] M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations,, Netw. Heterog. Media, 1 (2006), 295. [2] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41. [3] J. F. Bonnans and J. André, "Optimal Structure of Gas Transmission Trunklines,", Research Report available at Centre de recherche INRIA Saclay, (2009). [4] R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks of pipes and canals,, SIAM J. Control Optim., 48 (2009), 2032. doi: 10.1137/080716372. [5] R. M. Colombo, M. Herty and V. Sachers, On 2 $\times$ 2 conservation laws at a junction,, SIAM J. Math. Anal., 40 (2008), 605. doi: 10.1137/070690298. [6] 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. [7] M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives,, Adv. Model. Optim., 7 (2005), 9. [8] M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks,, ESAIM Control Optim. Calc. Var., (2009). [9] M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization,, Netw. Heterog. Media, 5 (2010), 299. doi: 10.3934/nhm.2010.5.299. [10] M. Herty, J. Mohring and V. Sachers, A new model for gas flow in pipe networks,, Math. Methods Appl. Sci., 33 (2010), 845. [11] M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks,, Netw. Heterog. Media, 2 (2007), 731. [12] 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. [13] T. Li, B. Rao and Z. Wang, Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions,, Discrete Contin. Dyn. Syst., 28 (2010), 243. doi: 10.3934/dcds.2010.28.243. [14] , Nord Stream AG,, www.nord-stream.com, (). [15] A. Osiadacz, "Simulation and Analysis of Gas Networks,", Gulf Publishing Company, (1987). [16] A. Osiadacz and M. Chaczykowski, "Comparison of Isothermal and Non-Isothermal Transient Models,", Technical Report available at Warsaw University of Technology, (1998). [17] A. Osiadacz and M. Chaczykowski, Comparison of isothermal and non-isothermal pipeline gas flow models,, Chemical Engineering J., 81 (2001), 41. doi: 10.1016/S1385-8947(00)00194-7. [18] , Pipeline Simulation Interest Group,, www.psig.org, (). [19] E. Sletfjerding and J. S. Gudmundsson, Friction factor in high pressure natural gas pipelines from roughness measurements,, International Gas Research Conference, (2001), 5. [20] M. C. Steinbach, On PDE solution in transient optimization of gas networks,, J. Comput. Appl. Math., 203 (2007), 345. doi: 10.1016/j.cam.2006.04.018. [21] Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems,, Chin. Ann. Math. Ser. B, 27 (2006), 643. doi: 10.1007/s11401-005-0520-2.

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##### References:
 [1] M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations,, Netw. Heterog. Media, 1 (2006), 295. [2] M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41. [3] J. F. Bonnans and J. André, "Optimal Structure of Gas Transmission Trunklines,", Research Report available at Centre de recherche INRIA Saclay, (2009). [4] R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks of pipes and canals,, SIAM J. Control Optim., 48 (2009), 2032. doi: 10.1137/080716372. [5] R. M. Colombo, M. Herty and V. Sachers, On 2 $\times$ 2 conservation laws at a junction,, SIAM J. Math. Anal., 40 (2008), 605. doi: 10.1137/070690298. [6] 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. [7] M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives,, Adv. Model. Optim., 7 (2005), 9. [8] M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks,, ESAIM Control Optim. Calc. Var., (2009). [9] M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization,, Netw. Heterog. Media, 5 (2010), 299. doi: 10.3934/nhm.2010.5.299. [10] M. Herty, J. Mohring and V. Sachers, A new model for gas flow in pipe networks,, Math. Methods Appl. Sci., 33 (2010), 845. [11] M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks,, Netw. Heterog. Media, 2 (2007), 731. [12] 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. [13] T. Li, B. Rao and Z. Wang, Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions,, Discrete Contin. Dyn. Syst., 28 (2010), 243. doi: 10.3934/dcds.2010.28.243. [14] , Nord Stream AG,, www.nord-stream.com, (). [15] A. Osiadacz, "Simulation and Analysis of Gas Networks,", Gulf Publishing Company, (1987). [16] A. Osiadacz and M. Chaczykowski, "Comparison of Isothermal and Non-Isothermal Transient Models,", Technical Report available at Warsaw University of Technology, (1998). [17] A. Osiadacz and M. Chaczykowski, Comparison of isothermal and non-isothermal pipeline gas flow models,, Chemical Engineering J., 81 (2001), 41. doi: 10.1016/S1385-8947(00)00194-7. [18] , Pipeline Simulation Interest Group,, www.psig.org, (). [19] E. Sletfjerding and J. S. Gudmundsson, Friction factor in high pressure natural gas pipelines from roughness measurements,, International Gas Research Conference, (2001), 5. [20] M. C. Steinbach, On PDE solution in transient optimization of gas networks,, J. Comput. Appl. Math., 203 (2007), 345. doi: 10.1016/j.cam.2006.04.018. [21] Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems,, Chin. Ann. Math. Ser. B, 27 (2006), 643. doi: 10.1007/s11401-005-0520-2.
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