Controllability properties of nonlinear systems induced by lifting

Abstract

Given a control system \(F\) with state space \(P\) and a control system \(f\) with state space \(M\), we define a notion of what it means for a mapping \(\varPhi :P\rightarrow M\) to lift the system \(f\) to \(F\). Roughly speaking, this means that trajectories of \(f\) can be associated to trajectories of \(F\) in such a manner that lifted trajectories of \(F\) are mapped by \(\varPhi \) back to the corresponding trajectories of \(f\). In this situation, we establish a fairly general sufficient condition for controllability properties of \(f\) to induce (or be lifted to) corresponding controllability properties for the system \(F\) via the mapping \(\varPhi \). We give two applications of this result. In the first application, \(F\) and \(f\) are linear systems and \(\varPhi \) is a linear mapping. In the second application, \(\varPhi \) defines a fiber-bundle structure on \(P\) over \(M\). These applications are in fact closely related to known results obtained by other researchers, but our objective is to show that these two seemingly disparate situations can be treated under the guise of a common framework. We also address an existence question: given a (controllable) system \(f\) on the state space \(M\) and a mapping \(\varPhi :P\rightarrow M\), when can the system \(f\) be lifted to a (controllable) system \(F\) on the state space \(P\) via the mapping \(\varPhi \)? Our final result shows that if \(\varPhi :P\rightarrow M\) is a \(k\)-fold covering space of \(M\) that lifts a system \(f\) on \(M\) to a system \(F\) on \(P\), then the global controllability of \(f\) implies the global controllability of \(F\).