We study the differentiability properties of the topological equivalence between a uniformly asymptotically stable linear nonautonomous system and a perturbed system with suitable nonlinearities. For this purpose, we construct a homeomorphism inspired in the Palmer's one restricted to the positive half line, studying additional continuity properties and providing sufficient conditions ensuring its Cr–smoothness.

We give necessary and sufficient conditions for a norm-compact subset of a Hilbert space to admit a ${{C}^{1}}$ embedding into a finite dimensional Euclidean space. Using quasibundles, we prove a structure theorem saying that the stratum of $n$-dimensional points is contained in an $n$-dimensional ${{C}^{1}}$ submanifold of the ambient Hilbert space. This work sharpens and extends earlier results of $\text{G}$. Glaeser on paratingents. As byproducts we obtain smoothing theorems for compact subsets of Hilbert space and disjunction theorems for locally compact subsets of Euclidean space.