Generic analytical embedding methods for nonstationary systems based on control theory

Abstract

We investigate by embedding methods a class of nonholonomic and forced systems arising in particular from the rolling pneumatic tire. For these systems the existing embedding theorems of Takens-type are not applicable due to the nonholonomic constraints, the time-dependence of the forces, and the noncompactness of the phase space. To overcome these difficulties we interpret the given mechanical system as a vector field on a nontrivial fiber bundle with time-dependent Riemannian fiber metric. I. INTRODUCTION Our main goal is to interprete a nonautonomous and constrained system as a lifted system on a suitable fiber bundle with nontrivial bundle metric and to construct for this system a generic embedding. Let us recall for the precise characterization of the bundle structure some basic definitions (((5), (4)). A -fiber bundle is given by a surjective (i.e. a map onto) submersion , where and are -manifolds. The submersion is assumed to be locally trivial, i.e. there exists a manifold such that, for each point , there exists a neighborhood of and a -diffeomorphism with We say that is the bundle space, is the base space, is the projection, is a bundle chart and is the typical fiber. For any the set is the fiber over . A -vector bundle is a -fiber bundle whose fibers have a vector space structure.