December 2, 2025Open Access

Composite Lyapunov Criteria for Stability and Convergence with Applications to Optimization Dynamics

Key Points

Convergence was attained through newly proposed composite Lyapunov criteria, exploring its potential for nonlinear systems.
The method resulted in observable stability metrics and associated convergence rates, enhancing practical applications.
Analysis indicates that the approach offers strict decay via differential inequalities, revealing significant advances in the field.
This framework emphasizes critical sets, supporting broader applicability in various optimization contexts, especially with minimal assumptions.

Abstract

We propose a composite Lyapunov framework for nonlinear autonomous systems that ensures strict decay through a pair of differential inequalities. The approach yields integral estimates, quantitative convergence rates, vanishing of dissipation measures, convergence to a critical set, and semistability under mild conditions, without relying on invariance principles or compactness assumptions. The framework unifies convergence to points and sets and is illustrated through applications to inertial gradient systems and Primal–Dual gradient flows.

Composite Lyapunov Criteria for Stability and Convergence with Applications to Optimization Dynamics

Key Points

Abstract

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