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

Published online by Cambridge University Press:  01 February 2023

Martin Gugat
Dynamics, Control and Numerics (Alexander von Humboldt–Professur), Friedrich–Alexander–Universität Erlangen–Nürnberg (FAU), Erlangen, Germany
Jens Habermann
Lehrstuhl für Partielle Differentialgleichungen, Friedrich–Alexander–Universität Erlangen– Nürnberg (FAU), Erlangen, Germany
Michael Hintermüller*
Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany Institut für Mathematik, Humboldt–Universität zu Berlin, Berlin, Germany
Olivier Huber
Institut für Mathematik, Humboldt–Universität zu Berlin, Berlin, Germany
*Correspondence author. Email:


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.

MSC classification

© The Author(s), 2023. Published by Cambridge University Press

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Banda, M., Herty, M. & Klar, A. (2006) Gas flow in pipeline networks. Networks Heterogen. Media 1, 4156.CrossRefGoogle Scholar
Brouwer, J., Gasser, I. & Herty, M. (2011) Gas pipeline models revisited: model hierarchies, nonisothermal models, and simulations of networks. Multiscale Model. Simul. 9, 601623.CrossRefGoogle Scholar
Fraunhofer, ISI, ISE & IEG (2021) Hydrogen metastudy – evaluation of energy system studies. Study commissioned by the National Hydrogen Council.Google Scholar
Freitas Rachid, F. B. & Costa Mattos, H. S. (1998) Modelling of pipeline integrity taking into account the fluid–structure interaction. Int. J. Numer. Methods Fluids 28, 337355.3.0.CO;2-6>CrossRefGoogle Scholar
Gillette, J. L. & Kolpa, R. L. (2008) Overview of interstate hydrogen pipeline systems. Technical Report, doi 10.2172/924391, Argonne National Lab. (ANL), Argonne, IL (USA).CrossRefGoogle Scholar
Gugat, M., Hante, F. M., Hirsch-Dick, M. & Leugering, G. (2015) Stationary states in gas networks. Networks Heterogen. Media 10, 295320.CrossRefGoogle Scholar
Gugat, M., Leugering, G., Martin, A., Schmidt, M., Sirvent, M. & Wintergerst, D. (2018) MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems. Comput. Optim. Appl. 70, 267294.CrossRefGoogle Scholar
Gugat, M. & Ulbrich, S. (2017) The isothermal Euler equations for ideal gas with source term: product solutions, flow reversal and no blow up. J. Math. Anal. Appl. 454, 439452.CrossRefGoogle Scholar
Gugat, M. & Ulbrich, S. (2018) Lipschitz solutions of initial boundary value problems for balance laws. Math. Models Methods Appl. Sci. 28, 921951.CrossRefGoogle Scholar
Gugat, M., Wintergerst, D. & Schultz, R. (2018) Networks of pipelines for gas with nonconstant compressibility factor: stationary states. Comput. Appl. Math. 37, 10661097.CrossRefGoogle Scholar
Gugat, M. & Zuazua, E. (2016) Exact penalization of terminal constraints for optimal control problems. Optim. Control Appl. Methods 37, 13291354.CrossRefGoogle Scholar
Grimm, V. (2021) Hydrogen as an opportunity for resilience and growth in Europe. In: V. Grimm, S. Nikutta et al. (editors), Deutschlands neue Agenda, Ullstein, Berlin.Google Scholar
Hante, F., Leugering, G., Martin, A., Schewe, L. & Schmidt, M. (2017) Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: from modeling to industrial applications. In: P. Manchanda, R. Lozi and A. H. Siddiqi (editors), Industrial Mathematics and Complex Systems: Emerging Mathematical Models, Methods and Algorithms, Springer Singapore, pp. 77122.CrossRefGoogle Scholar
Herty, M., Mohring, J. & Sachers, V. (2010) A new model for gas flow in pipe networks. Math. Meth. Appl. Sci. 33, 845855.CrossRefGoogle Scholar
Hintermüller, M. & Strogies, N. (2020) Identification of the friction coefficient in a semilinear system for gas transport through a network. Optim. Methods Software 35(3), 576617.CrossRefGoogle Scholar
Klamka, J. (2002) Constrained exact controllability of semilinear systems. Syst. Control Lett. 47, 139147.CrossRefGoogle Scholar
Kuczynski, S., Laciak, M., Olijnyk, A., Szurlej, A. & Wlodek, T. (2019) Thermodynamic and technical issues of hydrogen and Methane-Hydrogen mixtures pipeline transmission. Energies 12, 569. Scholar
Li, T.-T. & Zhang, B.-Y. (1998) Global exact controllability of a class of quasilinear hyperbolic systems. J. Math. Anal. Appl. 225, 289311.CrossRefGoogle Scholar
Osiadacz, A. J. (1996) Different Transient Flow Models – Limitations, Advantages, and Disadvantages. newblock PSIG-9606 Report. Pipeline Simulation Interest Group, San Francisco, California.Google Scholar
Sidki Uyar, T. & Besikci, D. (2017) Integration of hydrogen energy systems into renewable energy systems for better design of 100 percent renewable energy communities. Int. J. Hydrogen Energy 42, 24532456.CrossRefGoogle Scholar
Smit, R., Weeda, M. & de Groot, A. (2007) Hydrogen infrastructure development in The Netherlands. Int. J. Hydrogen Energy 32, 13871395.CrossRefGoogle Scholar
Witkowski, A., Rusin, A., Majkut, M. & Stolecka, K. (2017) Comprehensive analysis of hydrogen compression and pipeline transportation from thermodynamics and safety aspects. Energy 141, 25082518.CrossRefGoogle Scholar
Zou, G. P., Cheraghi, N. & Taheri, F. (2005) Fluid-induced vibration of composite natural gas pipelines. Int. J. Solids Struct. 42, 12531268.CrossRefGoogle Scholar