First-order differentiability of the flow of a system with L(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 the L(p)-topology. Roughly speaking, it is proved that, if the norm of the map f(t, x, u) defining the process together with its first derivatives goes to infinity, with the norm of u not faster than \\u\\(p), p > 1, then the flow is C-1 in the L(p)-topology. This property implies that, if the control maps are bounded, then the flow is differentiable in any L(p), p > 1. Moreover, it is proved that the only systems for which the flow is differentiable in L(1) are the affine ones.