# American Institute of Mathematical Sciences

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

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

Received  October 2010 Revised  January 2011 Published  June 2011

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.
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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