  The aim of this paper is to give a new description of the geometry appearing
in the multi-specialization along a general family of submanifolds of a real
analytic manifold (including some important cases as clean intersection or a
simultaneously linearizable family of Lagrangian submanifolds in a cotangent
bundle) and then, to extend several properties of the multi-specialization. The
notion of multi-asymptotic expansions is also extended. In the local model more
general cases are studied: locally we can construct new sheaves of
multi-asymptotically developable functions closely related with asymptotics
along a subvariety with a simple singularity such as a cusp.
