American Institute of Mathematical Sciences

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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

Martin Gugat 1, and  Markus Dick 1,

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

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 and 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), 295-314. 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), 41-56. doi: 10.3934/nhm.2006.1.41.

[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]

J.-M. Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 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), 2-11. 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), 691-709.

[8]

M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives, Adv. Model. Optim., 7 (2005), 9-37.

[9]

M. Gugat, Boundary feedback stabilization by time delay for one-dimensional wave equations, IMA J. Math. Control Inform., 27 (2010), 189-203. doi: 10.1093/imamci/dnq007.

[10]

M. Gugat, Stabilizing a vibrating string by time delay, in "15th International Conference on Methods and Models in Automation and Robotics (MMAR)," Miedzyzdroje, August 23-26, (2010), 144-147. 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), 28-51. 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), 745-757. doi: 10.1002/mma.1394.

[13]

M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization, Netw. Heterog. Media, 5 (2010), 299-314. 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), 845-855.

[15]

M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks, Netw. Heterog. Media, 2 (2007), 733-750. doi: 10.3934/nhm.2007.2.733.

[16]

T. Li, "Controllability and Observability for Quasilinear Hyperbolic Systems," AIMS Series on Applied Mathematics, 3, American Institute of Mathematical Sciences, Springfield, MO, Higher Education Press, Beijing, 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), 243-257. 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), 425-479. 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), 559-581. doi: 10.3934/dcdss.2009.2.559.

[21]

A. Osiadacz, "Simulation and Analysis of Gas Networks," Gulf Publishing Company, Houston, 1987.

[22]

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.

[23]

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.

[24]

J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim., 48 (2009), 2771-2797. 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), 517-554. doi: 10.1137/100796261.

[26]

Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems, Chinese Ann. Math. Ser. B, 27 (2006), 643-656. doi: 10.1007/s11401-005-0520-2.

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), 295-314. 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), 41-56. doi: 10.3934/nhm.2006.1.41.

[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]

J.-M. Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 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), 2-11. 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), 691-709.

[8]

M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives, Adv. Model. Optim., 7 (2005), 9-37.

[9]

M. Gugat, Boundary feedback stabilization by time delay for one-dimensional wave equations, IMA J. Math. Control Inform., 27 (2010), 189-203. doi: 10.1093/imamci/dnq007.

[10]

M. Gugat, Stabilizing a vibrating string by time delay, in "15th International Conference on Methods and Models in Automation and Robotics (MMAR)," Miedzyzdroje, August 23-26, (2010), 144-147. 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), 28-51. 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), 745-757. doi: 10.1002/mma.1394.

[13]

M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization, Netw. Heterog. Media, 5 (2010), 299-314. 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), 845-855.

[15]

M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks, Netw. Heterog. Media, 2 (2007), 733-750. doi: 10.3934/nhm.2007.2.733.

[16]

T. Li, "Controllability and Observability for Quasilinear Hyperbolic Systems," AIMS Series on Applied Mathematics, 3, American Institute of Mathematical Sciences, Springfield, MO, Higher Education Press, Beijing, 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), 243-257. 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), 425-479. 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), 559-581. doi: 10.3934/dcdss.2009.2.559.

[21]

A. Osiadacz, "Simulation and Analysis of Gas Networks," Gulf Publishing Company, Houston, 1987.

[22]

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.

[23]

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.

[24]

J. Valein and E. Zuazua, Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim., 48 (2009), 2771-2797. 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), 517-554. doi: 10.1137/100796261.

[26]

Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems, Chinese Ann. Math. Ser. B, 27 (2006), 643-656. doi: 10.1007/s11401-005-0520-2.

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