  This paper deals with the reconstruction of the desired demand in an optimal
control problem, stated over a tree-shaped transportation network which is
governed by a linear hyperbolic conservation law. As desired demands typically
undergo fluctuations due to seasonality or unexpected events making short-term
adjustments necessary, such an approach can exemplary be used for forecasting
from past data. We suggest to model this problem as a so-called inverse optimal
control problem, i.e., a hierarchical optimization problem whose inner problem
is the optimal control problem and whose outer problem is the reconstruction
problem. In order to guarantee the existence of solutions in the function space
framework, the hyperbolic conservation law is interpreted in weak sense
allowing for control functions in Lebesgue spaces. For the computational
treatment of the model, we transfer the hierarchical problem into a nonsmooth
single-level one by plugging the uniquely determined solution of the inner
optimal control problem into the outer reconstruction problem before applying
techniques from nonsmooth optimization. Some numerical experiments are
presented to visualize various features of the model including different types
of noise in the demand and strategies of how to observe the network in order to
obtain good reconstructions of the desired demand.
