Dynamics of aggregation and collapse in multidimensional swarming models

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

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.