Convergent realizations of Lie subalgebras

It is known since the seminal work of Guillemin and Sternberg that Lie subalgebras of finite codimension can be realized as subalgebras of formal vector fields over formal power series. In this note, we characterize the Lie subalgebras which admit a convergent realization in the sense of locally analytic vector fields. We give generalizations of these properties for the problem of output realization. We give reformulations and applications of these algebraic results in the context of control theory. In particular, we recover and clarify previous results on the realization of Chen-Fliess series for control-affine systems, the equivalence of control systems, the existence of embedded or canonical systems.