Optimal Control of Nonlinear Hyperbolic Conservation Laws by On/off-switching with an Application to Traffic Flow Modeling

This paper focuses on the differentiability properties of the control-to-state mapping for entropy solutions to a scalar hyperbolic conservation law on R with respect to the switching times of an on/off control. As an example we consider the traffic density on a unidirectional road described by the LWR-model that is controlled by a traffic light switching between red and green at certain times. These switching times are the control variables of the considered optimization problem, where a general tracking-type functional is minimized. We investigate the differentiability of the reduced objective function, also in the presence of shocks. We show that the state ypt̄, ̈q at some observation time t̄ depends differentiably on the switching times in a generalized sense that implies total differentiability for the composition with a tracking functional. Furthermore, we present an adjoint-based formula for the gradient of the reduced objective functional with respect to the switching times.