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 & 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, (2006). Google Scholar
[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. Google Scholar
[3]	M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41. Google Scholar
[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. doi: 10.3934/nhm.2009.4.189. Google Scholar
[5]	J. F. Bonnans and J. André, Optimal structure of gas transmission trunklines,, Research Report available at Centre de recherche INRIA Saclay, (2009). Google Scholar
[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. doi: 10.1137/080716372. Google Scholar
[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. doi: 10.1109/TAC.2006.887903. Google Scholar
[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. doi: 10.3934/nhm.2010.5.691. Google Scholar
[9]	M. Gugat, Optimal nodal control of networked hyperbolic systems: evaluation of derivatives,, Adv. Model. Optim., 7 (2005), 9. Google Scholar
[10]	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. Google Scholar
[11]	M. Gugat and M. Dick, Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction,, submitted., (). Google Scholar
[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. doi: 10.1051/cocv/2009035. Google Scholar
[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. Google Scholar
[14]	M. Herty, J. Mohring and V. Sachers, A new model for gas flow in pipe networks,, Math. Methods Appl. Sci., 33 (2010), 845. Google Scholar
[15]	M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks,, Netw. Heterog. Media, 2 (2007), 733. doi: doi:10.3934/nhm.2007.2.733. Google Scholar
[16]	T. Li, "Controllability and Observability for Quasilinear Hyperbolic Systems,", American Institute of Mathematical Sciences, (2010). Google Scholar
[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. Google Scholar
[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. doi: 10.3934/dcdss.2009.2.559. Google Scholar
[19]	A. Osiadacz, "Simulation and Analysis of Gas Networks,", Gulf Publishing Company, (1987). Google Scholar
[20]	A. Osiadacz and M. Chaczykowski, Comparison of isothermal and non-isothermal transient models,, Technical Report available at Warsaw University of Technology, (1998). Google Scholar
[21]	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. Google Scholar
[22]	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. Google Scholar
[23]	M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups,", Birkhäuser, (2009). doi: 10.1007/978-3-7643-8994-9. Google Scholar
[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. Google Scholar
[25]	Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems,, Chin. Ann. Math., 27B (2006), 643. doi: 10.1007/s11401-005-0520-2. Google Scholar

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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, (2006). Google Scholar
[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. Google Scholar
[3]	M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks,, Netw. Heterog. Media, 1 (2006), 41. Google Scholar
[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. doi: 10.3934/nhm.2009.4.189. Google Scholar
[5]	J. F. Bonnans and J. André, Optimal structure of gas transmission trunklines,, Research Report available at Centre de recherche INRIA Saclay, (2009). Google Scholar
[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. doi: 10.1137/080716372. Google Scholar
[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. doi: 10.1109/TAC.2006.887903. Google Scholar
[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. doi: 10.3934/nhm.2010.5.691. Google Scholar
[9]	M. Gugat, Optimal nodal control of networked hyperbolic systems: evaluation of derivatives,, Adv. Model. Optim., 7 (2005), 9. Google Scholar
[10]	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. Google Scholar
[11]	M. Gugat and M. Dick, Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction,, submitted., (). Google Scholar
[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. doi: 10.1051/cocv/2009035. Google Scholar
[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. Google Scholar
[14]	M. Herty, J. Mohring and V. Sachers, A new model for gas flow in pipe networks,, Math. Methods Appl. Sci., 33 (2010), 845. Google Scholar
[15]	M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks,, Netw. Heterog. Media, 2 (2007), 733. doi: doi:10.3934/nhm.2007.2.733. Google Scholar
[16]	T. Li, "Controllability and Observability for Quasilinear Hyperbolic Systems,", American Institute of Mathematical Sciences, (2010). Google Scholar
[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. Google Scholar
[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. doi: 10.3934/dcdss.2009.2.559. Google Scholar
[19]	A. Osiadacz, "Simulation and Analysis of Gas Networks,", Gulf Publishing Company, (1987). Google Scholar
[20]	A. Osiadacz and M. Chaczykowski, Comparison of isothermal and non-isothermal transient models,, Technical Report available at Warsaw University of Technology, (1998). Google Scholar
[21]	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. Google Scholar
[22]	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. Google Scholar
[23]	M. Tucsnak and G. Weiss, "Observation and Control for Operator Semigroups,", Birkhäuser, (2009). doi: 10.1007/978-3-7643-8994-9. Google Scholar
[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. Google Scholar
[25]	Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems,, Chin. Ann. Math., 27B (2006), 643. doi: 10.1007/s11401-005-0520-2. Google Scholar

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