Spatiotemporal Optimal Control of Reaction-Diffusion Malware Propagation in Multi-AUV Networks

The safety issues of multiple autonomous underwater vehicle (multi-AUV) networks are crucial but have been insufficiently addressed in open underwater environments, where threats like malware propagate across both space and time. This paper establishes a three-layer reaction-diffusion SIRS model to accurately describe the spatiotemporal cross-propagation evolution of malware among multi-AUVs at different layers, incorporating spatial heterogeneity and a spatial-dependent infection rate. The existence of an optimal control pair is established using minimization techniques, and the differentiability of the control-to-state mapping is demonstrated to derive the first-order optimality conditions. Two optimal spatiotemporal control measures are proposed to effectively inhibit malware propagation within highly infected areas while minimizing control costs. Finally, the proposed optimal control strategies have been shown to be effective through numerical simulations, as they dynamically suppress the spread of malware in both time and space while minimizing the control costs.