AI 中文总结
该研究将对齐模型结果扩展至根有向图，在更弱的持续激励条件下，证明一阶模型可渐近共识、二阶模型可渐近群集，说明弱通信下仍能涌现集体行为。
AI 中文摘要
本文研究具有非普适交互、时滞和可能通信故障的一阶与二阶对齐模型，将文献[17]的结果扩展至仅假设为根的交互有向图，并采用更弱的持续激励条件。特别地，我们允许固定长度时间区间内的交互量随时间多项式衰减。对于一阶Hegselmann-Krause型模型，我们在通信权重衰减指数与到根的最大距离相关的合适条件下，证明其渐近收敛至共识；对于二阶Cucker-Smale模型，在影响函数衰减的额外假设下，我们建立其渐近群集行为。这些结果表明，在通信逐渐减弱且无需交互图强连通的情况下，集体行为仍可涌现。
英文摘要
In this paper, we investigate first- and second-order alignment models with non-universal interaction, time delays and possible communication failures, extending the results in [17] to interaction digraphs that are only assumed to be rooted and to a weaker Persistence Excitation Condition. In particular, we allow the amount of interaction over time intervals of fixed length to decay polynomially in time. For the first-order Hegselmann-Krause type model, we prove asymptotic convergence to consensus under a suitable condition relating the decay exponent of the communication weights to the maximal distance from the root. For the second-order Cucker-Smale model, we establish asymptotic flocking under an additional assumption on the decay of the influence function. These results show that collective behavior can still emerge under progressively weakening communication and without requiring strong connectivity of the interaction graph.
