First-order differentiability of the flow of a system withL p controls

This paper considers the study of the regularity of the flow of a nonautonomous nonlinear control process when the set of control maps is endowed with theL^p-topology. Roughly speaking, it is proved that, if the norm of the mapf(t, x, u) defining the process together with its first derivatives goes to infinity, with the norm ofu not faster thanu^p,p>1, then the flow isC¹ in theL^p-topology. This property implies that, if the control maps are bounded, then the flow is differentiable in anyL^p,p>1. Moreover, it is proved that the only systems for which the flow is differentiable inL¹ are the affine ones.This research was supported by a grant from Ministero dell'Universitá e della Ricerca Scientifica e Tecnologica, Rome, Italy.