# Optimal Control of Switched Networks for Nonlinear Hyperbolic Conservation Laws

### From DFG-SPP 1253

## Project leader

Lehrstuhl für Angewandte Mathematik II

Uni Erlangen

Tel: 09131 85-67135

Fax: 09131 85-67134

Email: leugering@am.uni-erlangen.de

Homepage: http://www.am.uni-erlangen.de/home/leugering/index_de.htm

FB Mathematik, AG10

TU Darmstadt

Tel: 06151 16-2487

Fax: 06151 16-3185

Email: ulbrich@mathematik.tu-darmstadt.de

Homepage:

## DFG funded assistants

Sebastian Pfaff (TU Darmstadt)

pfaff@mathematik.tu-darmstadt.de

## Description

The aim of this project is the analytical study and numerical approximation of optimal control problems for networks of nonlinear hyperbolic conservation laws under modal switching. Such switching is considered at the boundary nodes and junctions of the network as well as in the source terms and fluxes, leading to a hybrid system of PDEs. Networks of this type arise for example in traffic flow models as well as in water and gas network models. Since entropy solutions of conservation laws may develop shocks, the analysis and numerical solution of control problems for conservation laws is difficult. Nevertheless, encouraging progress has been achieved recently by the applicants and others for the optimal control of conservation laws. Switching may lead to additional discontinuities in the solution, which is quite natural in the context of entropy solutions. On one hand a careful analysis of these optimal control problems shall be conducted, in particular existence of optimal controls, differentiability properties of the objective function with respect to controls (switching times, boundary controls, etc.), corresponding sensitivity and adjoint equations, optimality conditions. On the other hand the appropriate numerical discretization of optimal control problems for switched networks of conservation laws shall be considered. The project will start with networks of scalar conservation laws and then proceed to $2 \times 2$ -systems in one space dimension.