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.