# Convergence of stochastic nonlinear systems and implications for Stochastic Model Predictive Control

## TL;DR

This paper establishes conditions under which stochastic nonlinear systems converge with probability 1, providing insights into the stability and performance of stochastic Model Predictive Control (MPC) formulations.

## Contribution

It introduces an input-to-state stability framework to prove convergence of stochastic nonlinear systems and discusses implications for stochastic MPC stability and performance.

## Key findings

- Proves convergence with probability 1 for stochastic nonlinear systems.
- Derives conditions for stability of stochastic MPC.
- Analyzes implications for existing stochastic MPC formulations.

## Abstract

The stability of stochastic Model Predictive Control (MPC) subject to additive disturbances is often demonstrated in the literature by constructing Lyapunov-like inequalities that ensure closed-loop performance bounds and boundedness of the state, but tight ultimate bounds for the state and non-conservative performance bounds are typically not determined. In this work we use an input-to-state stability property to find conditions that imply convergence with probability 1 of a disturbed nonlinear system to a minimal robust positively invariant set. We discuss implications for the convergence of the state and control laws of stochastic MPC formulations, and we prove convergence results for several existing stochastic MPC formulations for linear and nonlinear systems.