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Constrained exact boundary controllability of a semilinear model for pipeline gas flow

Part of: Control systems Controllability, observability, and system structure

Published online by Cambridge University Press:  01 February 2023

Martin Gugat ,
Jens Habermann ,
Michael Hintermüller Open the ORCID record for Michael Hintermüller [Opens in a new window]  and
Olivier Huber
Martin Gugat
Affiliation:
Dynamics, Control and Numerics (Alexander von Humboldt–Professur), Friedrich–Alexander–Universität Erlangen–Nürnberg (FAU), Erlangen, Germany
Jens Habermann
Affiliation:
Lehrstuhl für Partielle Differentialgleichungen, Friedrich–Alexander–Universität Erlangen– Nürnberg (FAU), Erlangen, Germany
Michael Hintermüller*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany Institut für Mathematik, Humboldt–Universität zu Berlin, Berlin, Germany
Olivier Huber
Affiliation:
Institut für Mathematik, Humboldt–Universität zu Berlin, Berlin, Germany
*
*Correspondence author. Email: hintermueller@wias-berlin.de
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Abstract

While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.

Keywords

Gas pipeline flowexact controllabilitycontinuous solutionsconstrained exact controllabilityboundary controlmethod of characteristics

MSC classification

Primary: 93B05: Controllability
Secondary: 93C10: Nonlinear systems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Special Issue 3: The Mathematics in Renewable Energy special issue , June 2023 , pp. 532 - 553
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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