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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

Markus Dick 1, , Martin Gugat 1, and  Günter Leugering 1,

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.
Keywords: gun barrel, linear networks, gas networks, networked hyperbolic systems, feedback law, Lyapunov function, critical length., Riemann invariants, Classical solutions.
Mathematics Subject Classification: 76N25, 35L50, 93C2.
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.   Google Scholar

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41.   Google Scholar

[3]

J. F. Bonnans and J. André, "Optimal Structure of Gas Transmission Trunklines,", Research Report available at Centre de recherche INRIA Saclay, (2009).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives,, Adv. Model. Optim., 7 (2005), 9.   Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[10]

M. Herty, J. Mohring and V. Sachers, A new model for gas flow in pipe networks,, Math. Methods Appl. Sci., 33 (2010), 845.   Google Scholar

[11]

M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks,, Netw. Heterog. Media, 2 (2007), 731.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

, Nord Stream AG,, www.nord-stream.com, ().   Google Scholar

[15]

A. Osiadacz, "Simulation and Analysis of Gas Networks,", Gulf Publishing Company, (1987).   Google Scholar

[16]

A. Osiadacz and M. Chaczykowski, "Comparison of Isothermal and Non-Isothermal Transient Models,", Technical Report available at Warsaw University of Technology, (1998).   Google Scholar

[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.  Google Scholar

[18]

, Pipeline Simulation Interest Group,, www.psig.org, ().   Google Scholar

[19]

E. Sletfjerding and J. S. Gudmundsson, Friction factor in high pressure natural gas pipelines from roughness measurements,, International Gas Research Conference, (2001), 5.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.   Google Scholar

[2]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41.   Google Scholar

[3]

J. F. Bonnans and J. André, "Optimal Structure of Gas Transmission Trunklines,", Research Report available at Centre de recherche INRIA Saclay, (2009).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives,, Adv. Model. Optim., 7 (2005), 9.   Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[10]

M. Herty, J. Mohring and V. Sachers, A new model for gas flow in pipe networks,, Math. Methods Appl. Sci., 33 (2010), 845.   Google Scholar

[11]

M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks,, Netw. Heterog. Media, 2 (2007), 731.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

, Nord Stream AG,, www.nord-stream.com, ().   Google Scholar

[15]

A. Osiadacz, "Simulation and Analysis of Gas Networks,", Gulf Publishing Company, (1987).   Google Scholar

[16]

A. Osiadacz and M. Chaczykowski, "Comparison of Isothermal and Non-Isothermal Transient Models,", Technical Report available at Warsaw University of Technology, (1998).   Google Scholar

[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.  Google Scholar

[18]

, Pipeline Simulation Interest Group,, www.psig.org, ().   Google Scholar

[19]

E. Sletfjerding and J. S. Gudmundsson, Friction factor in high pressure natural gas pipelines from roughness measurements,, International Gas Research Conference, (2001), 5.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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