Time and Energy Costs for Flocking of Cucker–Smale System Under Denial-of-Service Attacks

This article investigates how Denial-of-Service (DoS) attacks impact the time and energy costs (ECs) associated with the emergence of flocking dynamics in the Cucker-Smale system. We propose resilient finite-time and fixed-time control protocols against DoS attacks and establish conditions under which the Cucker-Smale (C-S) system can achieve flocking within a bounded time. The attack patterns are modeled stochastically and are constrained by the effective duration of the attack. Explicit upper bounds for both time and ECs are derived, demonstrating their dependence not only on the group size and control parameters, but also on the duration of DoS attacks. Theoretical analysis and numerical simulations consistently demonstrate that shorter attack durations facilitate faster convergence and lower energy consumption. Additionally, our analysis uncovers a tradeoff between time cost and EC under DoS attacks, suggesting that optimal communication intensity should be carefully adjusted to meet specific performance requirements of practical applications.