On homotopy properties of solutions of some differential inclusions in the W 1, P -topology

We consider a differential inclusion on a manifold, defined by a field of open half-spaces whose boundary in each tangent space is the kernel of a one-form ω. We make the assumption that the corank one distribution associated to the kernel of ω is completely nonholonomic of step 2. We identify a subset of solutions of the differential inclusion, satisfying two endpoints and periodic boundary conditions, which are homotopy equivalent in the W1,p-topology, for any p ∈ [1,+∞), to the based loop space and the free loop space respectively.