Fixed-point stability conditions and convergence analysis for distributed autonomous decision-making system

Abstract

Distributed autonomous decision-making processes may be mathematically modeled as an ensemble of nonlinear systems with decision components that depend upon each other and whose states evolve through iteration using a mapping process. Establishing the existence, stability, and convergence of these distributed systems is still a challenging mathematical problem due to nonlinear interactions and disturbances. In this paper, a mathematical technique using a fixed-point theory framework is proposed for analyzing distributed nonlinear decision mappings using a combination of operator-theoretic analysis and iterative computation methods. The decision process is formulated as a nonlinear fixed-point operator equation, and sufficient conditions for stability are established based on the idea of contractive mappings in complete metric spaces. A distributed fixed point iterative framework with a relaxation-based update mechanism is proposed to enhance convergence reliability in the presence of coupled decision interactions. Computational experiments show that the presented mathematical model meets the developed stability conditions and achieves theoretical convergence behavior with lower approximation errors. These results show that the developed mathematical framework rigorously guarantees equilibrium analysis, convergence analysis, and stability of distributed nonlinear decision systems.