Abstract
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.
Highlights
We study the well-posedness of the corresponding time-discrete optimal control problem
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, on a single edge
The Euler equations are given by a system of nonlinear hyperbolic partial differential equations (PDEs) which represent the motion of a compressible non-viscous fluid or a gas
Summary
The Euler equations are given by a system of nonlinear hyperbolic partial differential equations (PDEs) which represent the motion of a compressible non-viscous fluid or a gas. They consist of the continuity equation, the balance of moments and the energy equation. Let ρ denote the density, v the velocity of the gas and p the pressure. In the subsonic case ( v < c ), the one which we consider in the sequel, two boundary conditions have to be imposed, one on the left and one on the right end of the pipe.
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