Calculus of Variations and Geometric Measure Theory

G. Buttazzo - A. Pratelli - S. Solimini - E. Stepanov

Optimal urban networks via mass transportation

created by stepanov on 21 Apr 2007
modified by pratelli on 25 Sep 2010


Published Paper

Inserted: 21 apr 2007
Last Updated: 25 sep 2010

Journal: Springer Lecture Notes in Mathematics
Volume: 1961
Pages: 160 pp.
Year: 2009

ISBN 3540857982, 9783540857983


The monograph is dedicated to a class of models of optimization of transportation networks (urban traffic networks or networks of railroads and highways) in the given geographic area. One assumes that the data on distributions of population and of servicesworkplaces (i.e. sources and sinks of the network) as well as the costs of movement with and without the help of the network to be constructed, are known. Further, the models take into consideration both the cost of everyday movement of the population and the cost of construction and maintenance of the network, the latter being determined by a given function on the total length of the network. The above data suffice, if one considers optimization in long-term prospective, while for the short-term optimization one also needs to know the transport plan of everyday movements of the population (i.e.\ the information on ``who goes where''). Similar models can also be adapted for the optimization of networks of different nature, like telecommunication, pipeline or drainage networks. In the monograph we study the most general problem settings, namely, when neither the shape nor even the topology of the network to be constructed is a priori known.

To be more precise, given a region $\Omega\subseteq*R*^N$, the transportation network to be constructed is modeled by an a priori generic Borel set $\Sigma\subseteq \Omega$. Then one considers the mass transportation problem in which the paths inside and outside the network $\Sigma$ are charged differently. The aim is to find the best location for $\Sigma$, in order to minimize a suitable cost functional ${\cal F}(\Sigma)$, which is given by the sum of the cost of transportation of the population, and the penalization term depending on the length of the network, which represents the cost of construction and maintenance of the network. To study the problem of existence of optimal solutions, first a relaxed version of the optimization problem, where the network is represented by a Borel measure rather than a set, is considered, and the existence of a relaxed solution is proven. One studies then the properties of optimal relaxed solutions (measures) and proves that, under suitable assumptions, the relaxed solutions solve the original problem, i.e. in fact they correspond to rectifiable sets, and therefore can be called classical solutions. However, it is shown that in general the problem studied may have no classical solutions. Furthermore, some topological properties of optimal networks, like closedness and the number of connected components, are studied. In particular, it is possible to find rather sharp conditions on problem data, which ensure the existence of closed optimal networks andor optimal networks having at most countably many connected components. Finally, a general regularity result on optimal networks is proven. Namely, it is shown that an optimal network is covered by a finite number of Lipschitz curves of uniformly bounded length, although it may have even uncountably many connected components.

Keywords: Optimal Networks, urban planning, Mass transportation