Dynamic Defense Strategies for Cyber-Physical Systems Using Stackelberg Games and Deep Reinforcement Learning in Discrete and Continuous Time

As cyber threats to power grid infrastructures escalate, the urgency of understanding how to protect cyber-physical systems (CPS) has never been greater. These systems, which integrate physical processes with digital control, are increasingly susceptible to sophisticated cyberattacks that can lead to widespread disruption. While most existing defense models function within either discrete or continuous-time frameworks, this research tackles a significant limitation: the absence of a unified strategy that encompasses both temporal domains.This study presents a hybrid defense framework that combines Stackelberg game theory with Deep Reinforcement Learning (DRL), aiming to provide flexible and adaptive protection. The objective of this framework is to facilitate proactive defense decisions that can anticipate and respond to attacks with strategic precision.We conducted extensive simulations using Python to assess the proposed model in both discrete and continuous time scenarios. Our approach was rigorously tested under realistic adversarial conditions to confirm its resilience and cost-effectiveness.Key findings indicate that defender-first strategies in discrete time effectively minimize system damage and alleviate computational burdens, while continuous-time responses, although immediate, demand significantly higher resource investment.This dual-domain solution offers a robust, adaptable toolset for real-time CPS defense. It supports infrastructure operators in navigating nonlinear, dynamic environments by combining theoretical rigor with practical impact—an urgently needed step in an increasingly complex threat landscape. Mathematics Subject Classification (2010) 68M10 · 91A23 · 93C10