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.
Intellectual Merit: 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. Results from this award focus on group dynamics that result when agents interact through potentials that depend only on pairwise distances. The research has many new results that focus on confining potentials in which the group size does not scale with the number of agents. Such potentials have been shown by the PI and collaborators to yield unusual patterns that could be exploited in automated control applications. This work focuses on analytical results for existence and stability of solutions. Broader Impacts: The work has involved training of PhD students, postdocs and undergraduates in advanced mathematical techniques in applied differential equations, numerical analysis and scientific computing. The theoretical results are now being used for engineering design of cooperative control platforms through collaborations with Engineering Schools at Arizona State University and Univ. Toledo.
