BECS: Collaborative Research: Characterization and Control of Emergent Behavior in Complex Systems

目前复杂网络系统领域的知识状况缺乏能够准确预测并帮助控制这类系统的行为的正式理论。该项目的科学目标是开发一种理论基础，以了解、分析和控制在不确定环境中运行的复杂系统的紧急行为。为了验证理论思想，该项目特别考虑了复杂的群体机器人系统，并旨在将所提出的分析框架应用于此类系统。中心方法使用随机微分方程组的数学框架，其中群被视为由非线性相互作用势耦合的耗散哈密顿系统。在这个框架中，不确定性(噪声)的影响被用从福克-普朗克方法获得的离散和连续公式进行数学处理。这项研究的重点是发展复杂系统建模的理论基础，并基于给定的控制输入和固有的系统不确定性进行分析。该框架便于对复杂系统中的复杂相互作用、高维、非线性和不确定性进行建模，以及对期望的紧急行为的控制。这项研究未来的潜在应用包括电网和通信网络。此外，该项目产生了更广泛的影响，包括促进教学、培训和学习；广泛传播以增进理解；以及让代表性不足的群体参与。本科生和研究生通过课程和参与研究和推广活动得到指导和培训。通过针对动态系统和控制以及数学界的会议上的特别会议、讲习班和教程，实现了广泛的传播。通过暑期项目、夏令营和其他项目，如女性参与科学与工程和新兴民族工程师，促进了本科生和来自代表性不足群体的学生的参与。

The current state of knowledge in the area of complex networked systems lacks formal theories that can accurately predict, and hence help control the behavior of such systems. The scientific objective of this project is to develop a theoretical basis to understand, analyze, and control emergent behavior in complex systems operating in uncertain environments. In order to validate the theoretical ideas, the project specifically considers complex swarm robotic systems and aims to apply the proposed analytical framework to such systems. The central approach uses the mathematical framework of stochastic differential equations wherein the swarm is treated as a dissipative Hamiltonian system coupled by nonlinear interacting potentials. In this framework, the influence of uncertainties (noise) is treated mathematically using both the discrete and continuum formulations to be obtained from a Fokker-Planck approach. The focus of this research is the development of a theoretical basis for modeling of complex systems and their analysis based upon given control inputs and inherent system uncertainty. This framework facilitates modeling the complex interactions, higher-dimensionality, nonlinearity, and uncertainty in complex systems as well as the control of desirable emergent behavior. Potential future applications of this research include power grids and communication networks. Furthermore, this project has broader impacts that include promoting teaching, training, and learning; broad dissemination to enhance understanding; and involvement of the underrepresented groups. Undergraduate and graduate students are mentored and trained via courses and involvement in research and outreach activities. Broad dissemination is achieved through special sessions, workshops, and tutorials at conferences that target both the dynamic systems and control as well as mathematics communities. Participation of undergraduates and students from underrepresented groups is facilitated via summer projects, summer camps, and other programs such as Women in Science and Engineering and Emerging Ethnic Engineers.