On homotopy properties of solutions of some differential inclusions in the $W^{1,p}$-topology
Erasmo Caponio, Antonio Masiello, Stefan Suhr
TL;DR
This work connects differential inclusions on a manifold, defined by a corank-one, step-$2$ nonholonomic distribution via a kernel one-form $\omega$, to the topology of loop spaces in the $W^{1,p}$ setting. By establishing that the endpoint map restricted to the relevant trajectory spaces is a Hurewicz fibration and showing the control space is contractible, the authors prove that trajectory spaces with endpoint or periodic conditions are homotopy equivalent to the standard based and free loop spaces for all $p\in[1,\infty)$. The key contribution is a non-smooth inverse function theorem approach (Clarke) to construct local cross-sections and to derive CW-type and homotopy type results for the loop-like trajectory spaces. This yields explicit relationships among the homotopy groups, e.g., $\pi_k(\Omega^p_{x,x})\cong\pi_{k+1}(M)$, and demonstrates that the trajectory spaces inherit the loop-space topology, with potential implications for sub-Riemannian geometry and control theory in Sobolev settings.
Abstract
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 $p\in [1,+\infty)$, to the based loop space and the free loop space respectively.