# Contraction analysis of nonlinear random dynamical systems

1 ASCOLA - Aspect and composition languages
LINA - Laboratoire d'Informatique de Nantes Atlantique, Département informatique - EMN, Inria Rennes – Bretagne Atlantique
Abstract : In order to bring contraction analysis into the very fruitful and topical fields of stochastic and Bayesian systems, we extend here the theory describes in \cite{Lohmiller98} to random differential equations. We propose new definitions of contraction (almost sure contraction and contraction in mean square) which allow to master the evolution of a stochastic system in two manners. The first one guarantees eventual exponential convergence of the system for almost all draws, whereas the other guarantees the exponential convergence in $L_2$ of to a unique trajectory. We then illustrate the relative simplicity of this extension by analyzing usual deterministic properties in the presence of noise. Specifically, we analyze stochastic gradient descent, impact of noise on oscillators synchronization and extensions of combination properties of contracting systems to the stochastic case. This is a first step towards combining the interesting and simplifying properties of contracting systems with the probabilistic approach.
Document type :
Reports

Cited literature [2 references]

https://hal.inria.fr/hal-00864079
Contributor : Nicolas Tabareau <>
Submitted on : Thursday, September 26, 2013 - 11:31:02 AM
Last modification on : Wednesday, December 5, 2018 - 1:22:15 AM
Long-term archiving on: : Friday, April 7, 2017 - 3:22:19 AM

### Files

RR-8368.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00864079, version 2
• ARXIV : 1309.5317

### Citation

Nicolas Tabareau, Jean-Jacques Slotine. Contraction analysis of nonlinear random dynamical systems. [Research Report] RR-8368, INRIA. 2013, pp.17. ⟨hal-00864079v2⟩

Record views