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. We make the assumption that the corank one distribution associated to the kernel 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 $W^{1,p}$-topology, for any $p\in [1,+\mathrm{\infty })$, to the based loop space and the free loop space respectively.