# Robust and structural ergodicity analysis of stochastic biomolecular networks involving synthetic antithetic integral controllers

## TL;DR

This paper develops exact algebraic and polynomial methods to verify ergodicity and output controllability in stochastic biomolecular networks with synthetic antithetic integral controllers, accounting for uncertainties.

## Contribution

It introduces robust, exact theoretical techniques for analyzing ergodicity and controllability in stochastic reaction networks with uncertainties, extending previous linear programming approaches.

## Key findings

- Exact algebraic methods verify ergodicity under uncertainty.
- Polynomial techniques confirm output controllability.
- Illustrative examples demonstrate the methods' effectiveness.

## Abstract

Ergodicity and output controllability have been shown to be fundamental concepts for the analysis and synthetic design of closed-loop stochastic reaction networks, as exemplified by the use of antithetic integral feedback controllers. In [Gupta, Briat & Khammash, PLoS Comput. Biol., 2014], some ergodicity and output controllability conditions for unimolecular and certain classes of bimolecular reaction networks were obtained and formulated through linear programs. To account for context dependence, these conditions were later extended in [Briat & Khammash, CDC, 2016] to reaction networks with uncertain rate parameters using simple and tractable, yet potentially conservative, methods. Here we develop some exact theoretical methods for verifying, in a robust setting, the original ergodicity and output controllability conditions based on algebraic and polynomial techniques. Some examples are given for illustration.