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A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction
1. | Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen |
References:
[1] |
H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization," Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia, 2006. |
[2] |
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. |
[3] |
M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. |
[4] |
N. Bedjaoui, E. Weyer and G. Bastin, Methods for the localization of a leak in open water channels, Netw. Heterog. Media, 4 (2009), 189-210.
doi: 10.3934/nhm.2009.4.189. |
[5] |
J. F. Bonnans and J. André, Optimal structure of gas transmission trunklines, Research Report available at Centre de recherche INRIA Saclay, January 7, 2009. |
[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.
doi: 10.1137/080716372. |
[7] |
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. |
[8] |
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.
doi: 10.3934/nhm.2010.5.691. |
[9] |
M. Gugat, Optimal nodal control of networked hyperbolic systems: evaluation of derivatives, Adv. Model. Optim., 7 (2005), 9-37. |
[10] |
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. |
[11] |
M. Gugat and M. Dick, Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction,, submitted., ().
|
[12] |
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. |
[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: doi:10.3934/nhm.2007.2.733. |
[16] |
T. Li, "Controllability and Observability for Quasilinear Hyperbolic Systems," American Institute of Mathematical Sciences, Springfield, 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] |
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. |
[19] |
A. Osiadacz, "Simulation and Analysis of Gas Networks," Gulf Publishing Company, Houston, 1987. |
[20] |
A. Osiadacz and M. Chaczykowski, Comparison of isothermal and non-isothermal transient models, Technical Report available at Warsaw University of Technology, 1998. |
[21] |
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. |
[22] |
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. |
[23] |
M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups," Birkhäuser, Basel - Boston - Berlin, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[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] |
Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems, Chin. Ann. Math., 27B (2006), 643-656.
doi: 10.1007/s11401-005-0520-2. |
show all references
References:
[1] |
H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization," Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia, 2006. |
[2] |
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. |
[3] |
M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. |
[4] |
N. Bedjaoui, E. Weyer and G. Bastin, Methods for the localization of a leak in open water channels, Netw. Heterog. Media, 4 (2009), 189-210.
doi: 10.3934/nhm.2009.4.189. |
[5] |
J. F. Bonnans and J. André, Optimal structure of gas transmission trunklines, Research Report available at Centre de recherche INRIA Saclay, January 7, 2009. |
[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.
doi: 10.1137/080716372. |
[7] |
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. |
[8] |
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.
doi: 10.3934/nhm.2010.5.691. |
[9] |
M. Gugat, Optimal nodal control of networked hyperbolic systems: evaluation of derivatives, Adv. Model. Optim., 7 (2005), 9-37. |
[10] |
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. |
[11] |
M. Gugat and M. Dick, Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction,, submitted., ().
|
[12] |
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. |
[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: doi:10.3934/nhm.2007.2.733. |
[16] |
T. Li, "Controllability and Observability for Quasilinear Hyperbolic Systems," American Institute of Mathematical Sciences, Springfield, 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] |
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. |
[19] |
A. Osiadacz, "Simulation and Analysis of Gas Networks," Gulf Publishing Company, Houston, 1987. |
[20] |
A. Osiadacz and M. Chaczykowski, Comparison of isothermal and non-isothermal transient models, Technical Report available at Warsaw University of Technology, 1998. |
[21] |
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. |
[22] |
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. |
[23] |
M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups," Birkhäuser, Basel - Boston - Berlin, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[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] |
Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems, Chin. Ann. Math., 27B (2006), 643-656.
doi: 10.1007/s11401-005-0520-2. |
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