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.
Mots clés : classical solution, networked hyperbolic systems, gas networks, feedback law, Lyapunov function
@article{COCV_2011__17_1_28_0,
author = {Gugat, Martin and Herty, Micha\"el},
title = {Existence of classical solutions and feedback stabilization for the flow in gas networks},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {28--51},
publisher = {EDP-Sciences},
volume = {17},
number = {1},
year = {2011},
doi = {10.1051/cocv/2009035},
mrnumber = {2775185},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2009035/}
}
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 SP - 28 EP - 51 VL - 17 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2009035/ DO - 10.1051/cocv/2009035 LA - en ID - COCV_2011__17_1_28_0 ER -
%0 Journal Article %A Gugat, Martin %A Herty, Michaël %T Existence of classical solutions and feedback stabilization for the flow in gas networks %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 28-51 %V 17 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2009035/ %R 10.1051/cocv/2009035 %G en %F COCV_2011__17_1_28_0
Gugat, Martin; Herty, Michaël. Existence of classical solutions and feedback stabilization for the flow in gas networks. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 28-51. doi : 10.1051/cocv/2009035. http://www.numdam.org/articles/10.1051/cocv/2009035/
[1] , and , Coupling conditions for gas networks governed by the isothermal Euler equations. Networks and Heterogenous Media 1 (2006) 295-314. | MR | Zbl
[2] , and , Gas flow in pipeline networks. Networks and Heterogenous Media 1 (2006) 41-56. | MR | Zbl
[3] , and , Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (2002) 1024-1065. | MR | Zbl
[4] , and , On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Networks and Heterogenous Media 4 (2009). | MR | Zbl
[5] , An explicit equation for friction factor in pipe. Ind. Eng. Chem. Fund. 18 (1979) 296-297.
[6] , , and , Optimal control in networks of pipes and canals. SIAM J. Control Optim. 48 (2009) 2032-2050. | MR | Zbl
[7] , Control and Nonlinearity, Mathematical Surveys and Monographs 136. AMS, Providence (2007). | MR | Zbl
[8] , and , A Lyapunov approach to control irrigation canals modeled by the Saint-Venant equations, in Proc. Eur. Control Conf., Karlsruhe, Germany (1999).
[9] , and , 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] , , and , 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] , and , A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans. Automat. Contr. 52 (2007) 2-11. | MR | Zbl
[12] , and , Dissipative boundary conditions for one dimensional nonlinear hyperbolic systems. SIAM J. Control Optim. 47 (2008) 1460-1498. | MR | Zbl
[13] , , , and , Boundary feedback control in networks of open channels. Automatica 39 (2003) 1365-1376. | MR | Zbl
[14] and , 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). | MR | Zbl
[15] and , Global boundary controllability of the de St. Venant equations between steady states. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003) 1-11. | Numdam | MR | Zbl
[16] and , Global boundary controllability of the Saint-Venant system for sloped canals with friction. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 257-270. | Numdam | MR | Zbl
[17] , and , Global controllability between steady supercritical flows in channel networks. Math. Meth. Appl. Sci. 27 (2004) 781-802. | MR | Zbl
[18] , Coupling conditions for networked systems of Euler equations. SIAM J. Sci. Comp. 30 (2007) 1596-1612. | MR | Zbl
[19] and , Adjoint calculus for optimization of gas networks. Networks and Heterogeneous Media 2 (2007) 733-750. | MR | Zbl
[20] and , On the modeling and stabilization of flows in networks of open canals. SIAM J. Control Optim. 41 (2002) 164-180. | MR | Zbl
[21] , 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. | MR | Zbl
[22] , Exact boundary controllability of unsteady flows in a network of open canals. Math. Nachr. 278 (2005) 310-329. | MR | Zbl
[23] and , Semi-global C2 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems. Chin. Ann. Math. B 22 (2001) 325-336. | MR | Zbl
[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. | MR | Zbl
[25] and , Global exact boundary controllability for first order quasilinear hyperbolic systems of diagonal form. Int. J. Dynamical Systems Differential Equations 1 (2007) 12-19. | MR | Zbl
[26] and , Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series V. Durham, NC, USA (1985). | MR | Zbl
[27] , and , Mixed integer models for the stationary case of gas network optimization. Math. Programming 105 (2006) 563-582. | MR | Zbl
[28] , Gas Pipeline Hydraulics. Taylor and Francis, Boca Raton (2005).
[29] , Simulation of transient flow in gas networks. Int. J. Numer. Meth. Fluids 4 (1984) 13-23. | Zbl
[30] , Simulation and Analysis of Gas Networks. Gulf Publishing Company, Houston (1987).
[31] , 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] , On PDE Solution in Transient Optimization of Gas Networks. Technical Report ZR-04-46, ZIB Berlin, Germany (2004). | Zbl
[34] , 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] , Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chin. Ann. Math. B 27 (2006) 643-656. | MR | Zbl
[36] , Fluid Mechanics. McGraw-Hill, New York, USA (2002). | Zbl
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