# Equivalent Stability Notions, Lyapunov Inequality, and Its Application in Discrete-Time Linear Systems with Stochastic Dynamics Determined by an i.i.d. Process

## TL;DR

This paper establishes the equivalence of different stability notions for discrete-time linear systems with i.i.d. stochastic dynamics, derives a Lyapunov inequality condition for stability, and demonstrates its application in stability analysis and synthesis.

## Contribution

It proves the equivalence of stability notions under i.i.d. assumptions and derives a Lyapunov inequality condition that can be solved as a linear matrix inequality.

## Key findings

- Equivalence of stability notions under i.i.d. stochastic dynamics.
- Lyapunov inequality condition for stability in necessary and sufficient form.
- Numerical examples demonstrating the approach's effectiveness.

## Abstract

This paper is concerned with stability analysis and synthesis for discrete-time linear systems with stochastic dynamics. Equivalence is first proved for three stability notions under some key assumptions on the randomness behind the systems. In particular, we use the assumption that the stochastic process determining the system dynamics is independent and identically distributed (i.i.d.) with respect to the discrete time. Then, a Lyapunov inequality condition is derived for stability in a necessary and sufficient sense. Although our Lyapunov inequality will involve decision variables contained in the expectation operation, an idea is provided to solve it as a standard linear matrix inequality; the idea also plays an important role in state feedback synthesis based on the Lyapunov inequality. Motivating numerical examples are further discussed as an application of our approach.