When does the evoluted set have negligible boundary?

The evoluted set at time T is the union of all images of an initial set A via the flow at times t ∈ (0, T). Its regularity is for interest for control, being the attainable set in several relevant examples.We first show that it is not sufficient for A to have negligible boundary (i.e. zero Lebesgue measure) to ensure that the evoluted set has negligible boundary too. Instead, we prove that such property holds when A is a C
^1,1
domain.