Collaborative Research: Structural Frameworks for Output Control of Continuum Ensemble Systems

集成控制是关于使用一个共同的控制输入来同时控制大量的动态系统（在极限情况下，连续体）。这些动态系统本身可以是网络化的多智能体系统。由于该框架本质上具有弹性和可扩展性，因此集成控制在科学和工程的各个学科中都有许多应用，从量子自旋系统到神经科学，紧急行为和多机器人控制。该领域的许多现有工作都集中在整体控制集成系统的问题上。然而，人们通常只对控制某个输出函数感兴趣，比如均值。在目前阶段，对集成系统的输出控制知之甚少。该项目旨在填补这一空白。该项目的目标是建立新的工具和结果，以引入连续统集成系统输出控制的框架。该研究计划将整合多个数学领域的思想，如控制论、图论、李论、泛函分析和动力系统。该项目的研究成果将推动所有这些研究领域的发展。对网络系统连续集成的关注使得输出控制框架适应于解决上述各种应用领域中的问题。该项目将围绕几个中心对象，即与集成系统相关的输出函数、单个系统共享的稀疏模式、参数化空间的几何形状和系统动力学类型。具体而言，pi将针对两类重要的输出函数制定两个互补的研究重点，并研究它们与其他对象的关系：(1)在第一个重点中，pi将引入并解决集成系统的新可控性概念，称为子轮廓可控性。粗略地说，一个整体系统的子轮廓可以被理解为单个系统的某些子状态的集合。除其他外，pi将旨在为控制子剖面建立必要和充分的条件，并开发用于测试的算法。(2)在第二个推力中，pi将研究集成系统中单个系统的输出函数。这种输出函数的典型例子包括平均矩和更一般的高矩。本文的研究成果将为理解连续统参数化空间上动态剖面的统计特性问题提供见解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Ensemble control is about using a common control input to simultaneously steer a large population (in the limit, a continuum) of dynamical systems. These dynamical systems can themselves be networked multi-agent systems. Because the framework is by nature resilient and scalable, ensemble control finds many applications across various disciplines in science and engineering, ranging from quantum spin systems to neuroscience, emergent behavior, and multi-robot control. Many existing works in this area have focused on the problem of controlling an ensemble system as a whole. However, one is often interested in only controlling a certain output function, such as the mean. At the current stage, little is known for output control of an ensemble system. This project aims to fill the gap. The goal of the project is to establish new tool and results to usher in a framework for output control of continuum ensemble systems. The research program will integrate ideas from multiple areas in mathematics, such as control theory, graph theory, Lie theory, functional analysis, and dynamical systems. The research findings of this project will thus push forward the development of all these research areas. The focus on continuum ensembles of networked systems makes the output control framework adaptable to address problems in various application domains as mentioned above.This project will revolve around several central objects, namely, the output function associated with an ensemble system, the sparsity pattern shared by the individual systems, the geometry of the parameterization space, and the type of system dynamics. Specifically, the PIs will formulate two complementary research thrusts targeting two important classes of output functions and investigate their relations with the other objects: (1) In the first thrust, the PIs will introduce and address a new controllability notion for ensemble systems, termed sub-profile controllability. Roughly speaking, a sub-profile of an ensemble system can be understood as the collection of certain substates of the individual systems. Amongst others, the PIs will aim to establish necessary and sufficient conditions for a sub-profile to be controllable and to develop algorithms for testing. (2) In the second thrust, the PIs will study output functions that integrate individual systems of an ensemble system. Typical examples of such output functions include the average and, more generally, higher moments. The research findings of this thrust will provide insights for the problem of understanding the statistical properties of dynamic profiles over continuum parametrization space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.