Synchronized tra c plans and stability of optima

The irrigation problem is the problem of nding an e cient way to transport a measure μ onto a measure μ−. By e cient, we mean that a structure that achieves the transport (which, following [3], we call tra c 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 tra c 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 [3]. Finally, we discuss the stability of minimizers with respect to speci c variations of the cost functional.