Controlled invariant distributions and differential properties

Abstract

The theory of (controlled) invariant (co-)distributions is reviewed, emphasizing the theory of liftings of vector fields, one-forms and (co-)distributions to the tangent and cotangent bundle. In particular, it is shown how invariant distributions can be equivalently described as invariant submanifolds of the tangent and cotangent bundle. This naturally leads to the notion of an invariant Lagrangian subbundle of the Whitney sum of tangent and cotangent bundle, which amounts to a special case of the central equation of contraction analysis. The interconnection of the prolongation of a nonlinear control system (living on the tangent bundle of the state space manifold) with its Hamiltonian extension (defined on the cotangent bundle) is shown to result in a differential Hamiltonian system. The invariant submanifolds of this differential Hamiltonian system corresponding to Lagrangian subbundles are seen to result in general differential Riccati and differential Lyapunov equations. The established framework thus yields a geometric underpinning of recent advances in contraction analysis and convergent dynamics.