Nonoverlapping domain decomposition and virtual controls for optimal control problems of p-type on metric graphs

Abstract

We consider nonoverlapping domain decomposition methods for p-type ordinary and partial differential equations and corresponding optimal control problems on metric graphs. As an exemplary context, we choose a doubly nonlinear p-parabolic model with p=32 which can be retrieved from the one-dimensional Euler system and that has come to be known as friction dominated flow in gas pipe networks. We introduce an optimal control problem for such systems on a metric graph, where the pipes are represented by the edges. We then apply the rolling horizon approach and obtain a sequence of static optimal control problems of p-type. The main interest of this survey article is to describe and utilize nonoverlapping domain decomposition procedures for such optimal control problems in the context of virtual controls which lead to a decomposition of the entire optimality system such that the decomposed system is itself the optimality system of an optimal control problem. This transfers the parallel iterative domain decomposition method into a sequence of parallel optimal control problems. We depart from the classical domain decomposition methods described by P.L. Lions (1990) and J.L. Lions and O. Pironneau (1998, 1999, 2000) and extend those to problems on metric graphs. We provide some numerical simulations.