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The irrigation problem is the problem of finding an efficient way to transport a mea- sure 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.

Synchronized traffic plans and stability of optima