SGER: THEORY AND APPLICATIONS OF ENSEMBLE CONTROL

在量子系统控制中的许多应用都涉及到使用相同的控制场来控制具有不同动力学的连续动力系统。这就形成了一类新的控制问题，我们称之为集成控制。我们建议对这类问题进行基础研究，并计划开发新的方法和适用定理，利用李代数和向量场的非交换性，来理解这类系统的可控性条件和最优控制。研究这些问题的动机来自对量子系综动力学的同时操纵，在许多情况下，系综的元素在表征系统动力学的参数中表现出变化。从我们的分析工作中获得的结果方法可以直接应用于核磁共振（NMR）光谱和成像（MRI）中最佳脉冲序列的设计，并应用于结构生物学，医学诊断和核磁共振量子计算。智力优势：PI引入了集成控制的概念，并提出了开发新的方法来研究集成控制系统的基本性质，包括可控性和最优控制。衍生的方法为各种磁共振应用中的脉冲序列设计提供了系统和革命性的方法。这项工作有望提高灵敏度，改进蛋白质核磁共振光谱和医学成像的实验。集成控制中出现的新的数学结构为控制和系统理论的新发展提供了良好的动力。更广泛的影响：PI的研究计划将推进最先进的数学控制理论方法。这些方法的应用范围超出了磁共振脉冲设计的范围。它们适用于涉及量子动力学相干控制的所有领域，包括相干光谱学（电子自旋共振（ESR）、激光光谱学等）和量子信息处理的各个领域。这项研究/教育计划将使本科生和研究生从理论和实践的角度接触到跨学科的知识。

Numerous applications in control of quantum systems involve controlling a continuum of dynamical systems with different dynamics by using the same control field. This forms a new class of control problems and we call them Ensemble Control. We propose a fundamental investigation of such problems and plan to develop new methods and applicable theorems, employing Lie algebras and the noncommutativity of vector fields, to understand the controllability conditions and optimal control of such systems. The motivation for looking into these problems comes from simultaneous manipulation of dynamics of quantum ensembles, where, in many cases, the elements of the ensemble show variations in the parameters that characterize the system dynamics. The resulting methodology obtained from our analytical work can be directly applied to the design of optimal pulse sequences in nuclear magnetic resonance (NMR) spectroscopy and imaging (MRI), with applications to structural biology, medical diagnosis, and NMR quantum computing.Intellectual Merit: The PI has introduced notion of ensemble control and proposes to develop new methods for investigating fundamental properties of ensemble control systems including controllability and optimal control. The derived methodology provides systematic and revolutionary approaches for pulse sequence design in various magnetic resonance applications. The work promises enhanced sensitivity and improved experiments in protein NMR spectroscopy and medical imaging. The new mathematical structures appearing in ensemble control are excellent motivation for new developmentsin control and systems theory.Broader Impacts: The PI's research plan will advance state-of-the-art methods of mathematical control theory. The applications of these methods extend beyond the scope of pulse design in magnetic resonance. They are applicable to all areas involving coherent control of quantum dynamics, including various fields of coherent spectroscopy (electron spin resonance (ESR), laser spectroscopy, etc.) as well as quantum information processing. The coupled research/education proposal will expose undergraduate and graduate students to interdisciplinary knowledge in both theoretical and practical viewpoints.