Dynamics of aggregation and collapse in multidimensional swarming models

这个项目为聚集模型发展了详细的数学理论。 这种模型涉及成对相互作用的潜力，并出现在生物群集模型和控制理论应用的协调组和共识的背景下。 这些模型也出现在其他环境中，如材料科学应用，包括颗粒流。 应用微分方程的研究涉及非线性偏微分方程的严格分析、连续和离散模型的数值模拟、渐近分析和建模。 感兴趣的具体问题包括：（a）详细了解在运动聚集情况下的崩溃动力学;（B）分数耗散在这种模型中的作用;（c）离散到连续极限;（d）分析大量极限下离散群的标度特性和行为。 工作还涉及相关的数学模型的犯罪热点在住宅区carearies.设计和分析的合作控制和算法的自主代理是一个活跃的研究领域，应用于危险区域的监视，周边巡逻，和控制的自动驾驶汽车的团队。 这些问题的一些想法可以来自生物群体的建模，这些生物群体提供了自然界中有多少代理可以无缝交互的极好例子，有时在相对短距离的大距离交互中。 许多著名的模型，这些分组经验表现出很大的复杂性。 这项研究计划是理解这些问题的基础。

This project develops detailed mathematical theory for aggregation models. Such models involve pair-wise interaction potentials and arise in the context of biological swarming models and control theory applications for coordinated groups and consensus. These models also appear in other contexts such as materials science applications including granular flow. The research in applied differential equations involves rigorous analysis of nonlinear partial differential equations, numerical simulation of continuum and discrete models, and asymptotic analysis and modeling. Specific problems of interest include (a) a detailed understanding of the dynamics of collapse in the case of kinematic aggregations; (b) the role of fractional dissipation in such models; (c) the discrete to continuum limit; and (d) analysis of scaling properties and behaviors of discrete swarms in the limit of large numbers. The work also involves related mathematical models for crime hotspots in residential burglaries.The design and analysis of cooperative control and algorithms for autonomous agents is an active area of research with application to surveillance of hazardous areas, perimeter patrol, and control of teams of autonomous vehicles. Some ideas for such problems can come from modeling of biological groups that provide excellent examples in nature of how many agents can interact seamlessly, sometimes over large distances with relatively short range interactions. Many well-known models for these groupings empirically exhibit a great deal of complexity. This research program is fundamental to the understanding of these problems.