A Lyapunov-Based Small-Gain Theorem for Fixed-Time ISS: Theory, Optimization, and Games

TL;DR

This paper introduces a Lyapunov-based small-gain theorem for fixed-time input-to-state stability in interconnected nonlinear systems, enabling systematic analysis and control design for fixed-time convergence in complex dynamical networks.

Contribution

It develops a new Lyapunov-based small-gain theorem for fixed-time ISS, extending stability analysis tools to interconnected systems with fixed-time guarantees.

Findings

The theorem guarantees fixed-time ISS for interconnected systems under nonlinear small-gain conditions.

Application to optimization and game problems demonstrates controllers achieving fixed-time convergence.

Numerical examples validate the effectiveness of the proposed stability and control methods.

Abstract

We develop a Lyapunov-based small-gain theorem for establishing fixed-time input-to-state stability (FxT-ISS) guarantees in interconnected nonlinear dynamical systems. The proposed framework considers interconnections in which each subsystem admits a FxT-ISS Lyapunov function, providing robustness with respect to external inputs. We show that, under an appropriate nonlinear small-gain condition, the overall interconnected system inherits the FxT-ISS property. In this sense, the proposed result complements existing Lyapunov-based smallgain theorems for asymptotic and finite-time stability, and enables a systematic analysis of interconnection structures exhibiting fixed-time stability. To illustrate the applicability of the theory, we study feedback-based optimization problems with time-varying cost functions, and Nash-equilibrium seeking for noncooperative games with nonlinear dynamical plants in the loop. For both problems, we present a class of non-smooth gradient or pseudogradient-based controllers that achieve fixed-time convergence without requiring time-scale separation and using real-time feedback. Numerical examples are provided to validate the theoretical findings.