  We provide an account for the existence and uniqueness of solutions to rough
differential equations under the framework of controlled rough paths. The case
when the driving path is $\beta$-H\"older continuous, for $\beta>1/3$, is
widely available in the literature. In its extension to the case when
$\beta\leqslant1/3,$ a main challenge and missing ingredient is to show that
controlled roughs paths are closed under composition with Lipschitz
transformations. Establishing such a property precisely, which has a strong
algebraic nature, is a main purpose of the present article.
