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Nonoverlapping domain decomposition for optimal control problems governed by semilinear models for gas ﬂow in networks

Leugering G., Martin A., Schmidt M., Sirvent M.

Control and Cybernetics

2017

Vol. 46, no. 3

191--225

We consider optimal control problems for gas ﬂow in pipeline networks. The equations of motion are taken to be represented by a ﬁrst-order system of hyperbolic semilinear equations derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a network and introduce a tailored time discretization thereof. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a nonoverlapping domain decomposition of the optimal control problem on the graph into local problems on smaller sub-graphs—ultimately on single edges. We prove convergence of the domain decomposition method on networks and study the wellposedness of the corresponding time-discrete optimal control problems. The point of the paper is that we establish virtual control problems on the decomposed subgraphs such that the corresponding optimality systems are in fact equal to the systems obtained via the domain decomposition of the entire optimality system.