# Existence of classical solutions and feedback stabilization for the flow in gas networks

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 1, page 28-51
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topGugat, Martin, and Herty, Michaël. "Existence of classical solutions and feedback stabilization for the flow in gas networks." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 28-51. <http://eudml.org/doc/272957>.

@article{Gugat2011,

abstract = {We consider the flow of gas through pipelines controlled by a compressor station. Under a subsonic flow assumption we prove the existence of classical solutions for a given finite time interval. The existence result is used to construct Riemannian feedback laws and to prove a stabilization result for a coupled system of gas pipes with a compressor station. We introduce a Lyapunov function and prove exponential decay with respect to the L2-norm.},

author = {Gugat, Martin, Herty, Michaël},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {classical solution; networked hyperbolic systems; gas networks; feedback law; Lyapunov function},

language = {eng},

number = {1},

pages = {28-51},

publisher = {EDP-Sciences},

title = {Existence of classical solutions and feedback stabilization for the flow in gas networks},

url = {http://eudml.org/doc/272957},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Gugat, Martin

AU - Herty, Michaël

TI - Existence of classical solutions and feedback stabilization for the flow in gas networks

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2011

PB - EDP-Sciences

VL - 17

IS - 1

SP - 28

EP - 51

AB - We consider the flow of gas through pipelines controlled by a compressor station. Under a subsonic flow assumption we prove the existence of classical solutions for a given finite time interval. The existence result is used to construct Riemannian feedback laws and to prove a stabilization result for a coupled system of gas pipes with a compressor station. We introduce a Lyapunov function and prove exponential decay with respect to the L2-norm.

LA - eng

KW - classical solution; networked hyperbolic systems; gas networks; feedback law; Lyapunov function

UR - http://eudml.org/doc/272957

ER -

## References

top- [1] M.K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations. Networks and Heterogenous Media1 (2006) 295–314. Zbl1109.76052MR2223073
- [2] M.K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks. Networks and Heterogenous Media1 (2006) 41–56. Zbl1108.76063MR2219276
- [3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim.30 (2002) 1024–1065. Zbl0786.93009MR1178650
- [4] G. Bastin, J.-M. Coron and B. d'Andrea-Novel, On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Networks and Heterogenous Media 4 (2009). Zbl1185.37175MR2525202
- [5] N.H. Chen, An explicit equation for friction factor in pipe. Ind. Eng. Chem. Fund.18 (1979) 296–297.
- [6] 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. Zbl1196.35135MR2516198
- [7] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs 136. AMS, Providence (2007). Zbl1140.93002MR2302744
- [8] J.-M. Coron, B. d'Andréa-Novel and G. Bastin, A Lyapunov approach to control irrigation canals modeled by the Saint-Venant equations, in Proc. Eur. Control Conf., Karlsruhe, Germany (1999).
- [9] J.-M. Coron, B. d'Andréa-Novel and G. Bastin, On boundary control design for quasi-linear hyperbolic systems with entropies as Lyapunov functions, in Proc. 41st IEEE Conf. Decision Control, Las Vegas, USA (2002).
- [10] J.-M. Coron, B. d'Andréa-Novel, G. Bastin and L. Moens, Boundary control for exact cancellation of boundary disturbances in hyperbolic systems of conservation laws, in Proc. 44st IEEE Conf. Decision Control, Seville, Spain (2005).
- [11] 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. Contr.52 (2007) 2–11. Zbl1283.93211MR2286756
- [12] 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. Zbl1172.35008MR2407024
- [13] J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa-Novel and G. Bastin, Boundary feedback control in networks of open channels. Automatica39 (2003) 1365–1376. Zbl1175.93108MR2141681
- [14] K. Ehrhardt and M. Steinbach, Nonlinear gas optimization in gas networks, in Modeling, Simulation and Optimization of Complex Processes, H.G. Bock, E. Kostina, H.X. Pu and R. Rannacher Eds., Springer Verlag, Berlin, Germany (2005). Zbl1069.90014MR2789748
- [15] M. Gugat and G. Leugering, Global boundary controllability of the de St. Venant equations between steady states. Ann. Inst. H. Poincaré Anal. Non Linéaire20 (2003) 1–11. Zbl1032.93030MR1958159
- [16] M. Gugat and G. Leugering, Global boundary controllability of the Saint-Venant system for sloped canals with friction. Ann. Inst. H. Poincaré Anal. Non Linéaire26 (2009) 257–270. Zbl1154.76009MR2483821
- [17] M. Gugat, G. Leugering and E.J.P.G. Schmidt, Global controllability between steady supercritical flows in channel networks. Math. Meth. Appl. Sci.27 (2004) 781–802. Zbl1047.93028MR2055319
- [18] M. Herty, Coupling conditions for networked systems of Euler equations. SIAM J. Sci. Comp.30 (2007) 1596–1612. Zbl1173.35080MR2398880
- [19] M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks. Networks and Heterogeneous Media2 (2007) 733–750. Zbl1141.76057MR2357766
- [20] G. Leugering and E.J.P.G. Schmidt, On the modeling and stabilization of flows in networks of open canals. SIAM J. Control Optim.41 (2002) 164–180. Zbl1024.76009MR1920161
- [21] T.-T. Li, Exact controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals. Math. Meth. Appl. Sci.27 (2004) 1089–1114. Zbl1151.93316MR2063097
- [22] T.-T. Li, Exact boundary controllability of unsteady flows in a network of open canals. Math. Nachr.278 (2005) 310–329. Zbl1066.93005MR2110532
- [23] T.-T. Li and Y. Jin, Semi-global C2 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems. Chin. Ann. Math. B22 (2001) 325–336. Zbl1005.35058MR1845753
- [24] T.-T. Li and B. Rao, [Exact boundary controllability of unsteady flows in a tree-like network of open canals]. C. R. Acad. Sci. Paris Ser. I 339 (2004) 867–872. Zbl1116.93311MR2111724
- [25] T.-T. Li and Z. Wang, Global exact boundary controllability for first order quasilinear hyperbolic systems of diagonal form. Int. J. Dynamical Systems Differential Equations1 (2007) 12–19. Zbl1168.35388MR2492242
- [26] T.-T. Li and W.-C. Yu, Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series V. Durham, NC, USA (1985). Zbl0627.35001MR823237
- [27] A. Martin, M. Möller and S. Moritz, Mixed integer models for the stationary case of gas network optimization. Math. Programming105 (2006) 563–582. Zbl1085.90035MR2190834
- [28] E. Menon, Gas Pipeline Hydraulics. Taylor and Francis, Boca Raton (2005).
- [29] A. Osiadacz, Simulation of transient flow in gas networks. Int. J. Numer. Meth. Fluids4 (1984) 13–23. Zbl0538.76066
- [30] A.J. Osciadacz, Simulation and Analysis of Gas Networks. Gulf Publishing Company, Houston (1987).
- [31] A.J. Osciadacz, Different Transient Models – Limitations, advantages and disadvantages, in 28th Annual Meeting of PSIG (Pipeline Simulation Interest Group), San Francisco, California, USA (1996).
- [32] Pipeline Simulation Interest Group, www.psig.org.
- [33] M. Steinbach, On PDE Solution in Transient Optimization of Gas Networks. Technical Report ZR-04-46, ZIB Berlin, Germany (2004). Zbl1119.93045
- [34] Z. Vostrý, Transient Optimization of gas transport and distribution, in Proceedings of the 2nd International Workshop SIMONE on Innovative Approaches to Modelling and Optimal Control of Large Scale Pipelines, Prague, Czech Republic (1993) 53–62.
- [35] Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chin. Ann. Math. B27 (2006) 643–656. Zbl1197.93062MR2273803
- [36] F.M. White, Fluid Mechanics. McGraw–Hill, New York, USA (2002). Zbl0471.76001

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.