# Asymptotic stability of controlled differential equations. Part I: Young integrals

## TL;DR

This paper develops a unified analytical framework to study the stability and attractors of controlled differential equations driven by rough paths with finite p-variation, focusing on Young integrals, and generalizes existing results on global attractors.

## Contribution

It introduces a novel approach to analyze stationary states of controlled differential equations driven by rough paths using random dynamical systems, extending previous work on attractors.

## Key findings

- Established conditions for the existence of global pullback attractors.
- Proved criteria for the attractor to be a singleton, ensuring pathwise convergence.
- Generalized recent results on the stability of controlled differential equations.

## Abstract

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.