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

Markus Dick; Martin Gugat; Günter Leugering; Markus Dick; Martin Gugat; Günter Leugering

doi:10.3934/nhm.2010.5.691

Networks and Heterogeneous Media

2010, Volume 5, Issue 4: 691-709. doi: 10.3934/nhm.2010.5.691

Previous Article Next Article

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

Received: 01 February 2010 Revised: 01 August 2010
76N25, 35L50, 93C20.

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.
- Classical solutions,
- networked hyperbolic systems,
- gas networks,
- linear networks,
- gun barrel,
- feedback law,
- Lyapunov function,
- Riemann invariants,
- critical length
Citation: Markus Dick, Martin Gugat, Günter Leugering. Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[J]. Networks and Heterogeneous Media, 2010, 5(4): 691-709. doi: 10.3934/nhm.2010.5.691

Related Papers:

Abstract

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.

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), 295-314.
[2]	M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.
[3]	J. F. Bonnans and J. André, "Optimal Structure of Gas Transmission Trunklines," Research Report available at Centre de recherche INRIA Saclay, January 7, 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), 2032-2050. 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), 605-622. 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), 2-11. doi: 10.1109/TAC.2006.887903
[7]	M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives, Adv. Model. Optim., 7 (2005), 9-37.
[8]	M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM Control Optim. Calc. Var., published online August 11, 2009.
[9]	M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization, Netw. Heterog. Media, 5 (2010), 299-314. 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), 845-855.
[11]	M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks, Netw. Heterog. Media, 2 (2007), 731-748.
[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), 164-180. 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), 243-257. doi: 10.3934/dcds.2010.28.243
[14]	www.nord-stream.com.
[15]	A. Osiadacz, "Simulation and Analysis of Gas Networks," Gulf Publishing Company, Houston, 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), 41-51. doi: 10.1016/S1385-8947(00)00194-7
[18]	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, Amsterdam, November 5-8, 2001.
[20]	M. C. Steinbach, On PDE solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361. 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), 643-656. doi: 10.1007/s11401-005-0520-2

Reader Comments

Your name:*

Email:*
© 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)