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

包围控制是关于使用一个共同的控制输入来同时操纵大量（在极限情况下，连续体）的动力系统。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，因此整体控制在科学和工程的各个学科中找到了许多应用，从量子自旋系统到神经科学、涌现行为和多机器人控制。在这一领域的许多现有的作品都集中在控制作为一个整体的合奏系统的问题。然而，人们通常只对控制某个输出函数感兴趣，例如均值。在目前阶段，很少有人知道的合奏系统的输出控制。本项目旨在填补这一差距。该项目的目标是建立新的工具和结果，以引入一个连续体集成系统的输出控制框架。该研究计划将整合来自数学多个领域的思想，如控制理论，图论，李理论，泛函分析和动力系统。本项目的研究成果将推动这些研究领域的发展。对连续集成网络系统的关注使得输出控制框架适用于解决上述各种应用领域中的问题。本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以及系统动力学类型。具体来说，PI将制定两个互补的研究重点，针对两个重要的输出函数类，并调查他们与其他对象的关系：（1）在第一个推力，PI将介绍和解决一个新的可控性概念，系综系统，称为子轮廓可控性。粗略地说，系综系统的子配置文件可以理解为单个系统的某些子状态的集合。除其他外，PI将旨在建立子配置文件可控的必要和充分条件，并开发测试算法。(2)在第二个重点中，PI将研究集成系综系统的各个系统的输出函数。这样的输出函数的典型示例包括平均矩，并且更一般地，包括更高的矩。该奖项反映了NSF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.