Dynamic Games with Mixed State-Control Constraints and Uncertain Mathematical Models: ε-Nash Equilibrium by DNN Realization

Uncertain dynamic games have recently emerged as a rigorous and versatile framework for the analysis and synthesis of multi-agent decision-making processes in complex, stochastic, and dynamically evolving environments. By integrating foundational concepts from dynamic game theory with neural network-based function approximation techniques, these methodologies facilitate the development of adaptive, data-driven strategies for agents whose interactions unfold over time and are subject to both state and control constraints. Notwithstanding these advances, practical implementations are invariably influenced by model inaccuracies, exogenous disturbances, and parametric uncertainties, all of which may substantially impair system performance and jeopardize stability if left unmitigated. In this context, the present study examines dynamic game formulations defined on perturbed and uncertain system models, explicitly incorporating state and control constraints, with the objective of ensuring robustness and reliability in both competitive and cooperative settings. We consider a broad class of nonlinear dynamic games characterized by system dynamics affected by unknown disturbances and uncertain parameters. Within this framework, Dynamic Neural Networks (DNNs) are employed to approximate feasible solutions to the associated robust control problem, thereby enabling the characterization of ε-Nash equilibria through learning mechanisms driven by worst-case trajectory realizations. A comprehensive theoretical analysis is developed to elucidate the effects of perturbations and uncertainties on equilibrium existence, convergence behavior, and closed-loop stability properties. Furthermore, sufficient conditions are established under which the neural learning dynamics ensure boundedness and convergence to approximate Nash or saddle-point equilibria, despite the presence of modeling imperfections. The proposed framework effectively synthesizes principles from robust control theory and learning-based game-theoretic approaches, yielding formal guarantees that are often absent in purely data-driven methodologies. Finally, numerical simulations conducted on representative dynamic game scenarios substantiate the efficacy of the proposed approach, demonstrating enhanced robustness relative to nominal neural game formulations. These findings contribute to the advancement of dependable dynamic game architectures, with potential applications spanning autonomous systems, robotics, and networked control systems operating under uncertainty.