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Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 2743251, 11 pages
Research Article

Exact Boundary Controllability for Free Traffic Flow with Lipschitz Continuous State

Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Cauerstraße 11, 91058 Erlangen, Germany

Received 3 December 2015; Accepted 21 March 2016

Academic Editor: Chaudry Masood Khalique

Copyright © 2016 Martin Gugat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider traffic flow governed by the LWR model. We show that a Lipschitz continuous initial density with free-flow and sufficiently small Lipschitz constant can be controlled exactly to an arbitrary constant free-flow density in finite time by a piecewise linear boundary control function that controls the density at the inflow boundary if the outflow boundary is absorbing. Moreover, this can be done in such a way that the generated state is Lipschitz continuous. Since the target states need not be close to the initial state, our result is a global exact controllability result. The Lipschitz constant of the generated state can be made arbitrarily small if the Lipschitz constant of the initial density is sufficiently small and the control time is sufficiently long. This is motivated by the idea that finite or even small Lipschitz constants are desirable in traffic flow since they might help to decrease the speed variation and lead to safer traffic.