# Optimal Control of Networks of Nonlinear Hyperbolic Conservation Laws

Sebastian Pfaff^{*} and Stefan Ulbrich

Hyperbolic balance laws arise in many different applications such as traffic modeling or fluid mechanics. The difficulty in the optimal control of hyperbolic conservation laws stems from the presence of moving discontinuities (shocks) in the entropy solution. This leads to the issue that the control-to-state mapping is not differentiable in the usual sense. For the case of the optimal control of an initial(-boundary) value problem on $\mathbb{R}$, or an interval $I=(a,b)$ respectively, the concept of shift-differentiability turned out to be very useful. Indeed, since the solution operator for these problems is shift-differentiable, its composition with a tracking-type functional can shown to be Fréchet-differentiable. In this talk we consider optimal control problems of scalar conservation laws on simple networks, namely networks consisting of only one junction. We analyze the generalized differentiability properties of the solution operator w.r.t. initial data and node condition. We also discuss the case where the controls are the switching times between different modes.

Mathematics Subject Classification: 49K20 35L65 49J50 35R05 35R02

Keywords: optimal control; scalar conservation law; network

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