Convergent realizations of Lie subalgebras

Abstract

It has been known since the seminal work of Guillemin and Sternberg that Lie subalgebras of finite codimension of an arbitrary real or complex Lie algebra 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, and the existence of embedded or canonical systems.