Abstract This paper explores the complex behavior of advanced persistent threat (APT) attacks, characterized by a dual threat: the sophisticated manipulation of adversarial disturbance inputs and the exacerbation of system vulnerabilities due to environmental uncertainties. To address these security concerns in large-scale multi-agent industrial cyber-physical systems (CPSs), we develop a decentralized control framework using mean-field game (MFG) theory with multiplicative noise in the dynamics. Our approach effectively tackles the scalability challenges inherent in large-scale environments while countering both intelligent adversarial disturbances and operational uncertainties. By designing resilient and robust decentralized controllers, we ensure system stability and convergence, even under worst-case disturbance inputs. We prove that the mean-field approximation accurately captures the system's collective behavior, and the proposed decentralized controllers achieve ∈-Nash equilibrium. Numerical experiments, inspired by the Ukraine power grid attack, demonstrate the effectiveness of the proposed control strategy.
Shen et al. (2025) studied this question.