Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction

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December 2011, 1(4): 469-491. doi: 10.3934/mcrf.2011.1.469

Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction

1.	Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany, Germany

Received December 2010 Revised June 2011 Published November 2011

Abstract
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We consider the isothermal Euler equations with friction that model the gas flow through pipes. We present a method of time-delayed boundary feedback stabilization to stabilize the isothermal Euler equations locally around a given stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For this system we introduce a Lyapunov function with delay terms and develop time-delayed boundary controls for which the Lyapunov function decays exponentially with time. We present the stabilization method for a single gas pipe and for a star-shaped network of pipes.

Keywords: delay, delay term, Boundary control, gas network, star-shaped network, Lyapunov function, Euler equations, Riemann invariants, time delay., feedback stabilization, $L^2$-norm, friction term, hyperbolic PDE, feedback with delay, conservation law.

Mathematics Subject Classification: 76N25, 35L50, 93C2.

Citation: Martin Gugat, Markus Dick. Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control & Related Fields, 2011, 1 (4) : 469-491. doi: 10.3934/mcrf.2011.1.469

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. doi: 10.3934/nhm.2006.1.295.
[2]	M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41. doi: 10.3934/nhm.2006.1.41.
[3]	J. F. Bonnans and J. André, Optimal structure of gas transmission trunklines,, Research Report, (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]	J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (2007).
[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. Dick, M. Gugat and G. Leugering, Classical solutions and feedback stabilization for the gas flow in a sequence of pipes,, Netw. Heterog. Media, 5 (2010), 691.
[8]	M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives,, Adv. Model. Optim., 7 (2005), 9.
[9]	M. Gugat, Boundary feedback stabilization by time delay for one-dimensional wave equations,, IMA J. Math. Control Inform., 27 (2010), 189. doi: 10.1093/imamci/dnq007.
[10]	M. Gugat, Stabilizing a vibrating string by time delay,, in, (2010), 23. doi: 10.1109/MMAR.2010.5587248.
[11]	M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks,, ESAIM Control Optim. Calc. Var., 17 (2011), 28. doi: 10.1051/cocv/2009035.
[12]	M. Gugat, M. Herty and V. Schleper, Flow control in gas networks: Exact controllability to a given demand,, Math. Methods Appl. Sci., 34 (2011), 745. doi: 10.1002/mma.1394.
[13]	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.
[14]	M. Herty, J. Mohring and V. Sachers, A new model for gas flow in pipe networks,, Math. Methods Appl. Sci., 33 (2010), 845.
[15]	M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks,, Netw. Heterog. Media, 2 (2007), 733. doi: 10.3934/nhm.2007.2.733.
[16]	T. Li, "Controllability and Observability for Quasilinear Hyperbolic Systems,", AIMS Series on Applied Mathematics, 3 (2010).
[17]	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.
[18]	A. Marigo, Entropic solutions for irrigation networks,, SIAM J. Appl. Math., 70 (): 1711. doi: 10.1137/09074783X.
[19]	S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, Netw. Heterog. Media, 2 (2007), 425. doi: 10.3934/nhm.2007.2.425.
[20]	S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559. doi: 10.3934/dcdss.2009.2.559.
[21]	A. Osiadacz, "Simulation and Analysis of Gas Networks,", Gulf Publishing Company, (1987).
[22]	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.
[23]	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.
[24]	J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks,, SIAM J. Control Optim., 48 (2009), 2771. doi: 10.1137/080733590.
[25]	J.-M. Wang, B.-Z. Guo and M. Krstic, Wave equation stabilization by delays equal to even multiples of the wave propagation time,, SIAM J. Control Optim., 49 (2011), 517. doi: 10.1137/100796261.
[26]	Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems,, Chinese 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. doi: 10.3934/nhm.2006.1.295.
[2]	M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41. doi: 10.3934/nhm.2006.1.41.
[3]	J. F. Bonnans and J. André, Optimal structure of gas transmission trunklines,, Research Report, (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]	J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (2007).
[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. Dick, M. Gugat and G. Leugering, Classical solutions and feedback stabilization for the gas flow in a sequence of pipes,, Netw. Heterog. Media, 5 (2010), 691.
[8]	M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives,, Adv. Model. Optim., 7 (2005), 9.
[9]	M. Gugat, Boundary feedback stabilization by time delay for one-dimensional wave equations,, IMA J. Math. Control Inform., 27 (2010), 189. doi: 10.1093/imamci/dnq007.
[10]	M. Gugat, Stabilizing a vibrating string by time delay,, in, (2010), 23. doi: 10.1109/MMAR.2010.5587248.
[11]	M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks,, ESAIM Control Optim. Calc. Var., 17 (2011), 28. doi: 10.1051/cocv/2009035.
[12]	M. Gugat, M. Herty and V. Schleper, Flow control in gas networks: Exact controllability to a given demand,, Math. Methods Appl. Sci., 34 (2011), 745. doi: 10.1002/mma.1394.
[13]	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.
[14]	M. Herty, J. Mohring and V. Sachers, A new model for gas flow in pipe networks,, Math. Methods Appl. Sci., 33 (2010), 845.
[15]	M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks,, Netw. Heterog. Media, 2 (2007), 733. doi: 10.3934/nhm.2007.2.733.
[16]	T. Li, "Controllability and Observability for Quasilinear Hyperbolic Systems,", AIMS Series on Applied Mathematics, 3 (2010).
[17]	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.
[18]	A. Marigo, Entropic solutions for irrigation networks,, SIAM J. Appl. Math., 70 (): 1711. doi: 10.1137/09074783X.
[19]	S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,, Netw. Heterog. Media, 2 (2007), 425. doi: 10.3934/nhm.2007.2.425.
[20]	S. Nicaise, J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559. doi: 10.3934/dcdss.2009.2.559.
[21]	A. Osiadacz, "Simulation and Analysis of Gas Networks,", Gulf Publishing Company, (1987).
[22]	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.
[23]	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.
[24]	J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks,, SIAM J. Control Optim., 48 (2009), 2771. doi: 10.1137/080733590.
[25]	J.-M. Wang, B.-Z. Guo and M. Krstic, Wave equation stabilization by delays equal to even multiples of the wave propagation time,, SIAM J. Control Optim., 49 (2011), 517. doi: 10.1137/100796261.
[26]	Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems,, Chinese Ann. Math. Ser. B, 27 (2006), 643. doi: 10.1007/s11401-005-0520-2.

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