We consider optimal control problems for the flow of gas in a pipe network.
The equations of motions are taken to be represented by a semi-linear model
derived from the fully nonlinear isothermal Euler gas equations. We formulate
an optimal control problem on a given network and introduce a time discretization
thereof. We then study the well-posedness of the corresponding
time-discrete optimal control problem. In order to further reduce the complexity,
we consider an instantaneous control strategy. The main part of the
paper is concerned with a non-overlapping domain decomposition of the
semi-linear elliptic optimal control problem on the graph into local problems
on a small part of the network, ultimately on a single edge.
Brouwer, J., Gasser, I. and Herty, M. (2011) Gas Pipeline Models Revisited: Model Hierarchies, Nonisothermal Models, and Simulations of Networks. Multiscale Modeling & Simulation, 9, 601-623. https://doi.org/10.1137/100813580
Gugat, M., Leugering, G., Martin, A., Schmidt, M., Sirvent, M. and Wintergerst, D. (2016) Towards Simulation Based Mixedinteger Optimization with Differential Equations. Submitted.
Gugat, M., Leugering, G., Martin, A., Schmidt, M., Sirvent, M. and Wintergerst, D. (2017) MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems. Submitted.
Hante, F., Leugering, G., Martin, A., Schewe, L. and 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: ISIAM-Proceedings, Springer- Verlag, Berlin.
Lagnese, J.E and Leugering, G. (2003) Domain Decomposition Methods in Optimal Control of Partial Differential Equations (Volume 148 of International Series of Numerical Mathematics). Birkh?user Verlag, Basel.
Roubicek, T. (2013) Nonlinear Partial Differential Equations with Applications (Volume 153 of International Series of Numerical Mathematics). 2nd Edition, Birkh?user-Basel, Basel. https://doi.org/10.1007/978-3-0348-0513-1
Kogut, P.I. and Leugering, G. (2011) Optimal Control Problems for Partial Differential Equations on Reticulated Domains, Systems and Control: Foundations and Applications. Springer, New York. https://doi.org/10.1007/978-0-8176-8149-4