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arXiv · 2512.21314

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

Michael Tang·Miroslav Krstic·Jorge Poveda
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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.

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Michael Tang, Miroslav Krstic, Jorge Poveda. 2026-06-23. A Lyapunov-Based Small-Gain Theorem for Fixed-Time ISS: Theory, Optimization, and Games. https://arxiv.org/abs/2512.21314

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