ABSTRACT
We consider a system of 1 - d-diffusion-convection equations on graphs as being representative of flow or transport problems in channel, pipeline or root-systems appearing in many applications. Such systems are subject to controls the effect of which one wants to optimize. In these notes we discuss the mathematical framework of typical optimal control problems of such systems on graphs. We focus on singular perturbations with respect to diffusion and on the numerical implementation,
2INTRODUCTION
We consider diffusion-convection (or sometimes advection) equations on a graph G consisting of a set of vertices (nodes, joints) and a set of edges which are one-dimensional manifolds represented as simple C2-curves in with arclength function pi, respectively. Functions defined on Oi are consequently expressed on the interval (0, l i ), with l i denoting the length of link i. l i has to be finite. We consider finite graphs and use the following notation: for each i I there are indices
such that i.e. the edge i starts at the node and ends at This convention turns G into a directed graph. Conversely, to each vertex ? J with we associate the set of edges starting
at that node, and the set ending there. The set then denotes the set of edges incident at ? J . Consequently, the edge-degree d J of the node υ J is given by d J := |I J |. On each of the edges we consider a scalar 1-d diffusionadvection equation of the following type
where u i (x, t) represents the state (concentration, velocity of a fluid etc.) at the location x and time is a diffusion parameter (considered small), a
(2.1)
convection or transport parameter, and is a reaction coefficient. The sign condition on β i is not restrictive, as, whenever β i changes sign only a finite number of times, we can introduce an edge at each zero of βi and reverse the orientation. Later, however, sign-conditions will apply, as we consider a “flow” through the network. We have denoted in (2.1) by a dot the time-derivative and by a prime the spatial derivative.
