On a Geometric Notion of Duality in Nonlinear Control Systems

Controllability and observability properties of control systems manifest on the tangent bundle and its dual, the cotangent bundle, respectively. Linear systems admit the exploitation of this duality via the formulation of an explicit dual system, which is controllable iff the original system is observable and vice versa. While in the context of linear systems the duality is understood well and facilitates the conversion of controller and observer design schemes, there are a number of interesting and complex intricacies when it comes to nonlinear systems. This article aims to extend the geometric relationship between the controllable subspace of a linear system and the unobservable subspace of its dual to the realm of nonlinear systems on smooth manifolds. To this end, we prove the local existence of a dynamic system that can, in a geometric sense, be considered dual to the original one.