ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - October 2008 - Volume 14, Issue 04  

Research Article

Synchronized traffic plans and stability of optima

Bernot, Marca1 and Figalli, Alessioa2

a1 UMPA, ENS Lyon, 46 Allée d'Italie, 69007 Lyon, France; mbernot@umpa.ens-lyon.fr

a2 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56100 Pisa, Italy; a.figalli@sns.it

Abstract

The irrigation problem is the problem of finding an efficient way to transport a measure μ+ onto a measure μ-. By efficient, we mean that a structure that achieves the transport (which, following [Bernot, Caselles and Morel, Publ. Mat. 49 (2005) 417–451], we call traffic plan) is better if it carries the mass in a grouped way rather than in a separate way. This is formalized by considering costs functionals that favorize this property. The aim of this paper is to introduce a dynamical cost functional on traffic plans that we argue to be more realistic. The existence of minimizers is proved in two ways: in some cases, we can deduce it from a classical semicontinuity argument; the other cases are treated by studying the link between our cost and the one introduced in [Bernot, Caselles and Morel, Publ. Mat. 49 (2005) 417–451]. Finally, we discuss the stability of minimizers with respect to specific variations of the cost functional.

(Received April 20 2007)

(Online publication January 30 2008)

Key Words:

  • Irrigation problem;
  • traffic plans;
  • dynamical cost;
  • stability

Mathematics Subject Classification:

  • 49Q20;
  • 90B10;
  • 90B06;
  • 90B20