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

¹ Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Bari, Italy
² Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany

^* Corresponding author: erasmo.caponio@poliba.it

Received: 16 April 2024
Accepted: 24 December 2024

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 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 W^1,p-topology, for any p ∈ [1,+∞), to the based loop space and the free loop space respectively.