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.
