A strict H^1-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction

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2011, 1(2): 225-244. doi: 10.3934/naco.2011.1.225

A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction

Markus Dick ^1, , Martin Gugat ^1, and Günter Leugering ^1,

1.	Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen

Received October 2010 Revised January 2011 Published June 2011

Abstract
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We study the isothermal Euler equations with friction and consider non-stationary solutions locally around a stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For the corresponding initial-boundary value problem we prove the existence of a continuously differentiable solution and present a method of boundary feedback stabilization. We introduce a Lyapunov function which is a weighted and squared $H^1$-norm of the difference between the non-stationary and the stationary state. We develop boundary feedback conditions which guarantee that the Lyapunov function and the $H^1$-norm of the difference between the non-stationary and the stationary state decay exponentially with time. This allows us also to prove exponential estimates for the $C^0$- and $C^1$-norm.

Keywords: conservation law, feedback stabilization, $H^1$-norm, exponential stability, distributed parameter system, gas network, Riemann invariants., Lyapunov function, friction term, $C^1$-norm, feedback law, $C^0$-norm, isothermal Euler equations, Boundary control.

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

Citation: Markus Dick, Martin Gugat, Günter Leugering. A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 225-244. doi: 10.3934/naco.2011.1.225

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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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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