American Institute of Mathematical Sciences

Advanced
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

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, Philadelphia, 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), 295-314.  Google Scholar

[3]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.  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), 189-210. 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, January 7, 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), 2032-2050. 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), 2-11. 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), 691-709. 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), 9-37.  Google Scholar

[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.  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), 28-51. 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), 299-314. 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), 845-855.  Google Scholar

[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.  Google Scholar

[16]

T. Li, "Controllability and Observability for Quasilinear Hyperbolic Systems," American Institute of Mathematical Sciences, Springfield, 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), 243-257. 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), 559-581. doi: 10.3934/dcdss.2009.2.559.  Google Scholar

[19]

A. Osiadacz, "Simulation and Analysis of Gas Networks," Gulf Publishing Company, Houston, 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), 41-51. 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), 345-361. doi: 10.1016/j.cam.2006.04.018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  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), 295-314.  Google Scholar

[3]

M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56.  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), 189-210. 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, January 7, 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), 2032-2050. 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), 2-11. 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), 691-709. 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), 9-37.  Google Scholar

[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.  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), 28-51. 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), 299-314. 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), 845-855.  Google Scholar

[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.  Google Scholar

[16]

T. Li, "Controllability and Observability for Quasilinear Hyperbolic Systems," American Institute of Mathematical Sciences, Springfield, 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), 243-257. 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), 559-581. doi: 10.3934/dcdss.2009.2.559.  Google Scholar

[19]

A. Osiadacz, "Simulation and Analysis of Gas Networks," Gulf Publishing Company, Houston, 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), 41-51. 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), 345-361. doi: 10.1016/j.cam.2006.04.018.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Martin Gugat, Markus Dick. Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control & Related Fields, 2011, 1 (4) : 469-491. doi: 10.3934/mcrf.2011.1.469
[2]

Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97
[3]

Wael Bahsoun, Benoît Saussol. Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6657-6668. doi: 10.3934/dcds.2016089
[4]

Duo Wang, Zheng-Fen Jin, Youlin Shang. A penalty decomposition method for nuclear norm minimization with l1 norm fidelity term. Evolution Equations & Control Theory, 2019, 8 (4) : 695-708. doi: 10.3934/eect.2019034
[5]

Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015
[6]

P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151
[7]

Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907
[8]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Robustness of time-dependent attractors in H1-norm for nonlocal problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1011-1036. doi: 10.3934/dcdsb.2018140
[9]

Mikko Salo. Stability for solutions of wave equations with $C^{1,1}$ coefficients. Inverse Problems & Imaging, 2007, 1 (3) : 537-556. doi: 10.3934/ipi.2007.1.537
[10]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5449-5463. doi: 10.3934/dcdsb.2020353
[11]

Abdallah Benabdallah, Mohsen Dlala. Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021092
[12]

Yaru Xie, Genqi Xu. The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. Networks & Heterogeneous Media, 2016, 11 (3) : 527-543. doi: 10.3934/nhm.2016008
[13]

Pia Heins, Michael Moeller, Martin Burger. Locally sparse reconstruction using the $l^{1,\infty}$-norm. Inverse Problems & Imaging, 2015, 9 (4) : 1093-1137. doi: 10.3934/ipi.2015.9.1093
[14]

Ahmad Mousavi, Zheming Gao, Lanshan Han, Alvin Lim. Quadratic surface support vector machine with L1 norm regularization. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021046
[15]

Ben Green, Terence Tao, Tamar Ziegler. An inverse theorem for the Gowers $U^{s+1}[N]$-norm. Electronic Research Announcements, 2011, 18: 69-90. doi: 10.3934/era.2011.18.69
[16]

Min Zhao, Changzheng Qu. The two-component Novikov-type systems with peaked solutions and $ H^1 $-conservation law. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2857-2883. doi: 10.3934/cpaa.2020245
[17]

Tobias Breiten, Karl Kunisch. Boundary feedback stabilization of the monodomain equations. Mathematical Control & Related Fields, 2017, 7 (3) : 369-391. doi: 10.3934/mcrf.2017013
[18]

Ambroise Vest. On the structural properties of an efficient feedback law. Evolution Equations & Control Theory, 2013, 2 (3) : 543-556. doi: 10.3934/eect.2013.2.543
[19]

Jiangang Qi, Bing Xie. Extremum estimates of the $ L^1 $-norm of weights for eigenvalue problems of vibrating string equations based on critical equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3505-3516. doi: 10.3934/dcdsb.2020243
[20]

Shui-Hung Hou, Qing-Xu Yan. Nonlinear locally distributed feedback stabilization. Journal of Industrial & Management Optimization, 2008, 4 (1) : 67-79. doi: 10.3934/jimo.2008.4.67

 Impact Factor: 

Tools

Metrics

  • PDF downloads (124)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences