*Published Paper*

**Inserted:** 21 apr 2007

**Last Updated:** 25 sep 2010

**Journal:** Springer Lecture Notes in Mathematics

**Volume:** 1961

**Pages:** 160 pp.

**Year:** 2009

**Notes:**

ISBN 3540857982, 9783540857983

**Abstract:**

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
services*workplaces (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 and*or 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

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