Instantaneous Optimal Control of Friction Dominated Flow in a Gas-Network

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 nonlinear model derived from a semi-linear approximation of 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. This involves a p-Laplace-type problem on the graph with $$p=\frac {3}{2}$$ . We prove well-posedness, existence of optimal controls and derive a first order optimality condition.