Improved stability of optimal traffic paths

Abstract

Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. Given a flow (traffic path) that transports a given measure µ − onto a target measure µ + , along a 1dimensional network, the transportation cost per unit length is supposed in these models to be proportional to a concave power α ∈ (0, 1) of the intensity of the flow. In this paper we address an open problem in the book Optimal transportation networks by Bernot, Caselles and Morel and we improve the stability for optimal traffic paths in the Euclidean space R d , with respect to variations of the given measures (µ − , µ + ), which was known up to now only for α > 1 − 1 d . We prove it for exponents α > 1 − 1 d−1 (in particular, for every α ∈ (0, 1) when d = 2), for a fairly large class of measures µ + and µ − .

On the well-posedness of branched transportation

“…In this case, a simple argument relies on the fact that the minimal transport energy W α (ν n , ν) metrizes the weak- * -convergence of probability measures ν n * ⇀ ν (see [2,Lemma 6.11 and Proposition 6.12]). This property is false for α ≤ 1 − 1 /d, as shown in [17]. The threshold α = 1− 1 /d appears also because for α above this value any two probability measures with compact support in R d can be connected with finite cost.…”

“…Strategy of the proof. In analogy with previous works [1,2,17], to prove Theorem 1.1 we assume by contradiction that T is not optimal, denote T opt a minimizer, and we construct a better competitor for T n (n large enough) by "sewing" a small portion of the traffic path T n with a large portion of T opt . In the following we shortly describe some of the main ideas and difficulties behind the proof of Theorem 1.1.…”

“…A technical difficulty for our construction is related to the fact that, although the limit of the Lagrangian descriptions of T n provides a Lagrangian description of T , the latter could contain cycles and cancellations at the level of currents. This issue did not appear in [17,Theorem 1.2] because there the convergence T n * ⇀ T was not necessary to obtain a cheap connection of the slices. To overcome this and obtain a lower semicontinuity result which keeps track in the limit of those Lagrangian trajectories which have opposite orientations and therefore they would cancel at the Eulerian level, we employ some ideas from the theory of currents with coefficients in normed groups (see §3.9).…”

Stability of optimal traffic plans in the irrigation problem

“…When T = T P we say that P decomposes T . Following [5], a good decomposition, first introduced by Smirnov (see [19,Section 1.2]) for normal currents, is a decomposition where neither cycles nor cancellations occur: Definition 2.2 (Good decomposition). Let T and P be a transport path and traffic plan such that T = T P .…”

“…This problem may be cast in two main statical frameworks: an Eulerian one [20], based on vector valued measures (more precisely normal 1-currents) called transport paths, and a Lagrangian one [12,2], based on positive measures on a set of curves (or trajectories) called traffic plans. We refer to the book [3] for the general theory of branched transport, and to the first sections of the more recent works [16,5,8,6] and the references therein.…”