Controlled rough paths: a general Hopf-algebraic setting

We set up controlled rough paths for a class of combinatorial Hopf algebras, encompassing shuffle, Butcher-Connes-Kreimer and Munthe-Kaas--Wright Hopf algebras. The class of controls we consider encompasses both H\"older continuous paths and (not necessarily continuous) paths with bounded $p$-variation. We prove existence and uniqueness of the solution of a lifted initial value problem in this general setting by applying the fixed point method in a suitable Banach space of controlled rough paths, and we prove a universal limit theorem addressing the robustness of the solution with respect to the parameters and the initial condition.