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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 |
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). Google Scholar |
[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). Google Scholar |
[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. |
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.
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). Google Scholar |
[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). Google Scholar |
[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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