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@article{faucris.112463164,
abstract = {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 nonstationary 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.},
author = {Hirsch-Dick, Markus and Gugat, Martin and Leugering, Günter},
doi = {10.3934/naco.2011.1.225},
faupublication = {yes},
journal = {Numerical Algebra, Control and Optimization},
keywords = {Boundary control; C; C; Conservation law; Distributed parameter system; Exponential stability; Feedback law; Feedback stabilization; Friction term; Gas network; H; Isothermal Euler equations; Lyapunov function; Riemann invariants},
note = {UnivIS-Import:2015-03-09:Pub.2011.nat.dma.zentr.astric},
pages = {225-244},
peerreviewed = {Yes},
title = {{A} strict {H1}-{Lyapunov} function and feedback stabilization for the isothermal {Euler} equations with friction},
url = {http://www.aimsciences.org/journals/displayReferences.jsp?paperID=6330},
volume = {1},
year = {2011}
}
@article{faucris.112244264,
abstract = {We consider the subcritical flow in gas networks consisting of a finite linear sequence of pipes coupled by compressor stations. Such networks are important for the transportation of natural gas over large distances to ensure sustained gas supply. We analyse the system dynamics in terms of Riemann invariants and study stationary solutions as well as classical non-stationary solutions for a given finite time interval. Furthermore, we construct feedback laws to stabilize the system locally around a given stationary state. To do so we use a Lyapunov function and prove exponential decay with respect to the L^{2}-norm. © American Institute of Mathematical Sciences.},
author = {Hirsch-Dick, Markus and Gugat, Martin and Leugering, Günter},
doi = {10.3934/nhm.2010.5.691},
faupublication = {yes},
journal = {Networks and Heterogeneous Media},
keywords = {Classical solutions; Critical length; Feedback law; Gas networks; Gun barrel; Linear networks; Lyapunov function; Networked hyperbolic systems; Riemann invariants},
note = {UnivIS-Import:2015-03-09:Pub.2010.nat.dma.zentr.classi},
pages = {691-709},
peerreviewed = {Yes},
title = {{Classical} solutions and feedback stabilization for the gas flow in a sequence of pipes},
url = {http://aimsciences.org/journals/displayArticles.jsp?paperID=5645},
volume = {5},
year = {2010}
}
@inproceedings{faucris.118501504,
abstract = {We consider the feedback stabilization of quasilinear hyperbolic systems on star-shaped networks. We present boundary feedback controls with varying delays. The delays are given by C^{1}-functions with bounded derivatives. We obtain the existence of unique C^{1}-solutions on a given finite time interval. In order to measure the system evolution, we introduce an L^{2}-Lyapunov function with delay terms. The feedback controls yield the exponential decay of the Lyapunov function with time. This implies the exponential stability of the system. Our results can be applied on the stabilization of the isothermal Euler equations with friction that model the gas flow in pipe networks. © 2012 IEEE.},
author = {Hirsch-Dick, Markus and Gugat, Martin and Leugering, Günter},
booktitle = {Methods and Models in Automation and Robotics (MMAR)},
date = {2012-08-27/2012-08-30},
doi = {10.1109/MMAR.2012.6347931},
faupublication = {yes},
keywords = {Couplings; Delay; Feedback control; Lyapunov methods; Mathematical model; Propagation;},
note = {UnivIS-Import:2015-04-16:Pub.2012.nat.dma.lama1.feedba},
pages = {125-130},
peerreviewed = {Yes},
series = {IEEEXplore},
title = {{Feedback} stabilization of quasilinear hyperbolic systems with varying delays},
url = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6347931&isnumber=6347808},
venue = {Miedzyzdroje, Poland},
year = {2012}
}
@article{faucris.123311144,
abstract = {We analyze the subcritical gas flow through fan-shaped networks of pipes, that is, through tree-shaped networks with exactly one node where more than two pipes meet. The gas flow in pipe networks is modeled by the isothermal Euler equations, a hyperbolic PDE system of balance laws. For this system we analyze stationary states and classical nonstationary solutions locally around a stationary state on a finite time interval. Furthermore, we present a Lyapunov function and boundary feedback laws to stabilize a fan-shaped network around a given stationary state. © 2011 Society for Industrial and Applied Mathematics.},
author = {Gugat, Martin and Hirsch-Dick, Markus and Leugering, Günter},
doi = {10.1137/100799824},
faupublication = {yes},
journal = {SIAM Journal on Control and Optimization},
keywords = {Classical solutions; Critical length; Fan-shaped networks; Feedback law; Gas networks; Junction; Lyapunov function; Networked hyperbolic systems; Riemann invariants},
note = {UnivIS-Import:2015-03-09:Pub.2011.nat.dma.zentr.gasflo},
pages = {2101-2117},
peerreviewed = {Yes},
title = {{Gas} {Flow} in {Fan}-{Shaped} {Networks}: {Classical} {Solutions} and {Feedback} {Stabilization}},
url = {http://epubs.siam.org/sicon/resource/1/sjcodc/v49/i5/p2101{\_}s1},
volume = {49},
year = {2011}
}
@article{faucris.117458704,
abstract = {We consider the isothermal Euler equations without friction that simulate gas flow through a pipe. We consider the problem of boundary stabilisation of this system locally around a given stationary state. We present a feedback law that is linear in the physical variables and yields exponential decay of the system state. For the numerical solution of hyperbolic systems of conservation laws, the Jin-Xin relaxation scheme can be used. Therefore, we also consider the boundary stabilisation of the relaxation system by the linear Riemann feedback and present numerical examples that show the rapid exponential decay of the stabilised system. © 2012 Taylor & Francis Group, LLC.},
author = {Hirsch-Dick, Markus and Gugat, Martin and Herty, Michael and Steffensen, Sonja},
doi = {10.1080/00207179.2012.703787},
faupublication = {yes},
journal = {International Journal of Control},
keywords = {boundary feedback stabilisation; conservation laws; isothermal Euler equations; Lyapunov function; relaxation scheme},
note = {UnivIS-Import:2015-03-09:Pub.2012.nat.dma.zentr.onther},
pages = {1766-1778},
peerreviewed = {Yes},
title = {{On} the relaxation approximation of boundary control of the isothermal {Euler} equations},
volume = {85},
year = {2012}
}
@inproceedings{faucris.118588184,
abstract = {We consider the subcritical gas flow through star-shaped pipe networks. The gas flow is modeled by the isothermal Euler equations with friction. We stabilize the isothermal Euler equations locally around a given stationary state on a finite time interval. For the stabilization we apply boundary feedback controls with time-varying delays. The delays are given by C ^{1}-functions with bounded derivatives. In order to analyze the system evolution, we introduce an L ^{2}-Lyapunov function with delay terms. The boundary controls guarantee the exponential decay of the Lyapunov function with time. © 2013 IFIP International Federation for Information Processing.},
address = {Berlin, Heidelberg},
author = {Gugat, Martin and Dick, Markus and Leugering, Günter},
booktitle = {System Modeling and Optimization},
date = {2011-09-12/2011-09-16},
doi = {10.1007/978-3-642-36062-6{\_}26},
faupublication = {yes},
keywords = {boundary feedback stabilization; Euler equations; gas network; Lyapunov function; star-shaped network; time-varying delay},
note = {UnivIS-Import:2015-04-16:Pub.2013.nat.dma.zentr.stabil},
pages = {255-265},
peerreviewed = {Yes},
publisher = {Springer Verlag},
series = {IFIP Advances in Information and Communication Technology},
title = {{Stabilization} of the {Gas} {Flow} in {Star}-{Shaped} {Networks} by {Feedback} {Controls} with {Varying} {Delay}},
url = {http://link.springer.com/chapter/10.1007/978-3-642-36062-6{\_}26},
venue = {Berlin},
volume = {391},
year = {2013}
}
@article{faucris.115130224,
abstract = {Pipeline networks for gas transportation often contain circles. For such networks it is more difficult to determine the stationary states than for networks without circles. We present a method that allows to compute the stationary states for subsonic pipe flow governed by the isothermal Euler equations for certain pipeline networks that contain circles. We also show that suitably chosen boundary data determine the stationary states uniquely. The construction is based upon novel explicit representations of the stationary states on single pipes for the cases with zero slope and with nonzero slope. In the case with zero slope, the state can be represented using the Lambert-W function.},
author = {Gugat, Martin and Hante, Falk and Hirsch-Dick, Markus and Leugering, Günter},
doi = {10.3934/nhm.2015.10.295},
faupublication = {yes},
journal = {Networks and Heterogeneous Media},
keywords = {Network;isothermal Euler equations;stationary state;circles;cycles;gas transportation network},
pages = {295-320},
peerreviewed = {Yes},
title = {{STATIONARY} {STATES} {IN} {GAS} {NETWORKS}},
volume = {10},
year = {2015}
}
@article{faucris.109756064,
abstract = {We consider the isothermal Euler equations with friction that model the gas flow through pipes. We present a method of time-delayed boundary feedback stabilization to stabilize the isothermal Euler equations locally around a given stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For this system we introduce a Lyapunov function with delay terms and develop time-delayed boundary controls for which the Lyapunov function decays exponentially with time. We present the stabilization method for a single gas pipe and for a star-shaped network of pipes.},
author = {Gugat, Martin and Hirsch-Dick, Markus},
doi = {10.3934/mcrf.2011.1.469},
faupublication = {yes},
journal = {Mathematical Control and Related Fields},
keywords = {Boundary control; Conservation law; Delay; Delay term; Euler equations; Feedback stabilization; Feedback with delay; Friction term; Gas network; Hyperbolic PDE; L; Lyapunov function; Riemann invariants; Star-shaped network; Time delay},
note = {UnivIS-Import:2015-03-09:Pub.2011.nat.dma.zentr.timede},
pages = {469-491},
peerreviewed = {Yes},
title = {{Time}-delayed boundary feedback stabilization of the isothermal {Euler} equations with friction},
url = {http://aimsciences.org/journals/contentsListnew.jsp?pubID=483},
volume = {1},
year = {2011}
}