$({\text{Ad}}_{f,g} )$, $({\text{ad}}_{f,g} )$ and Locally $({\text{ad}}_{f,g} )$ Invariant and Controllability Distributions

In the study of nonlinear control systems, the concepts of an invariant foliation and an invariant distribution play important roles. In this paper we explore various forms of these concepts and show how they occur in the study of controllability, observability and decoupling of nonlinear systems.