  We provide a unified analytic approach to study stationary states of
controlled differential equations driven by rough paths, using the framework of
random dynamical systems and random attractors. Part I deals with driving paths
of finite $p$-variations with $1 \leq p <2$ so that the integrals are
interpreted in the Young sense. Our method helps to generalize recent results
\cite{GAKLBSch2010}, \cite{ducGANSch18}, \cite{duchongcong18} on the existence
of the global pullback attractors for the generated random dynamical systems.
We also prove sufficient conditions for the attractor to be a singleton, thus
the pathwise convergence is in both pullback and forward senses.
