  We introduce the class of controlled Loewner-Kufarev equations and consider
aspects of their algebraic nature. We lift the solution of such a controlled
equation to the (Sato)-Segal-Wilson Grassmannian, and discuss its relation with
the tau-function. We briefly highlight relations of the Grunsky matrix with
integrable systems and conformal field theory. Our main result is the explicit
formula which expresses the solution of the controlled equation in terms of the
signature of the driving function through the action of words in generators of
the Witt algebra.
