Boundedness of solutions in feedback systems with antithetic controllers

TL;DR

This paper proves that solutions of a class of nonlinear feedback systems with antithetic controllers in synthetic biology are inherently bounded over time, ensuring stability through a simple feedback mechanism.

Contribution

The paper demonstrates boundedness of solutions in nonlinear feedback systems with delayed antithetic controllers using a novel differential inequality approach without Lyapunov functions.

Findings

All system trajectories are bounded over time.

The feedback mechanism prevents unbounded growth of the state.

Boundedness is established via a transparent differential inequality method.

Abstract

This paper studies whether solutions of a class of nonlinear feedback systems remain bounded over time. The systems we consider arise naturally in synthetic biology, where the antithetic feedback controller regulates a biological process through a delayed feedback loop. Our main result is that every trajectory of such a system is bounded. The key insight is simple: if the regulated state grows too large for too long, the feedback loop will eventually respond and push it back down. More precisely, we show that whenever the state exceeds a threshold and remains there long enough, the feedback signal becomes strong enough to force the state to decrease. We then show that once this happens, the feedback remains strong enough to keep the state from growing unbounded. The proof works directly with differential inequalities and does not require constructing a Lyapunov function, making the mechanism transparent and easy to interpret. The boundedness result can be understood as a time-domain small-gain effect, where the delayed feedback ultimately counteracts any persistent growth in the system.