Groups and Singularities in Quantum Control and Nonlinear Design

该项目旨在开发i)量子控制中的状态准备的数值廉价方法；以及ii)存在奇点的非线性系统的控制律，以及奇异外部微分系统的技术。一些现有的和未来的技术要求控制量子系统--半导体异质结构、光谱学、原子激光、量子信息处理、分子动力学控制等等。在明智的建模和优化之后，这些问题中的许多可以通过作用于量子系统的外部控制被重塑为准备所需的么正发电机的问题。这很快就引出了李群上漂移的系统的路径规划问题，通过受约束的控制。具有漂移、输入少且受限的系统的路径规划到目前为止还没有被研究过。控制量子系统的唯一其他选择是计算代价高昂的优化，以及严重依赖直觉的跟踪。在这个项目中，直觉和最优化的作用将局限于重新制定一个给定的物理目标，作为量子系统的状态准备。后一个问题将通过发展酉阵的结构化因式分解来解决，其结构由约束决定。所得到的控制设计将与其他技术，如逆和最优控制进行比较。然后将在无反转激光、分子控制、量子计算和信息方面进行应用。相关问题，例如量子系统可达集合的易处理描述，也将被研究。项目的第二部分源于这样一个事实，即现有的非线性控制方案很少直接适用于奇异情况，如变秩解耦矩阵和相对度的变化。奇点理论、层理理论、非光滑分析和李群等工具将被用来解决这个问题。一个关键特征将是使用微分形式而不是向量场来研究这类问题。对于奇异的外微分系统，这也应该有回报。我们将从这个角度来研究诸如偏微分方程组和变型方程的中间积分法等技术的应用。值得一提的是，该项目的两个组成部分都需要类似的数学计算。该项目的成功完成应该会对当前和未来的几项技术产生好处。例如，制造量子计算机需要解决该项目中正在研究的问题的确切类型。同样，在分子水平上获得化学反应的快照也需要对量子过程的控制。对量子现象的主动控制也将导致通过调谐激光控制化学反应。这应该会提高化学反应中的产品专一性，从而最大限度地减少不需要的产品的存在。对量子现象的控制仍处于初级阶段。该项目旨在通过将其与传统控制理论具体联系起来，进一步推动它的发展。相反，该项目的结果应该有助于揭示宏观世界系统的控制。这对机器人、消费电子和电力网络等传统技术也有好处，因为基本的数学原理非常相似。从教学的角度来看，该项目的贡献将是向本科生和研究生表明，高等数学不是一个干旱的领域，而是一个活生生的过程，与现实生活情景有明显的联系。这应该会增加学生，特别是来自代表人数不足的群体的学生接受数学、科学和技术方面的高等教育的可能性。

0072415RamakrishnaThe project aims to develop i) numerically inexpensive methods for state preparation in quantum control; and ii) control laws for nonlinear systems in the presence of singularities, and techniques for singular exterior differential systems. Several extant and future technologies call for the control of quantum systems - semiconductor heterostructures, spectroscopy, atomic lasers, quantum information processing, control of molecular dynamics, to name a few. Many of these problems can, after judicious modeling and optimization, be recast as the problem of preparing desired unitary generators, via an external control acting on a quantum system. This leads quickly to the problem of path planning for systems with drift on Lie groups, via controls that are constrained. Path planning for systems with drift, with few and constrained inputs, has not hitherto been studied. The only other alternatives for the control of quantum systems are computationally expensive optimization, and tracking which relies heavily on intuition. In this project the role of intuition and optimization will be limited to reformulating a given physical objective as state preparation for a quantum system. The latter problem will be attacked by developing structured factorization of unitary matrices, the structure being determined by the constraints. The resulting control design will be compared with other techniques such as inversion and optimal control. Applications to lasing without inversion, molecular control, quantum computing and information will be then carried out. Related issues, such as tractable descriptions of reachable sets of quantum systems, will also be studied. The second part of the project stems from the fact that the extant non-linear control schemes are rarely directly applicable in singular situations such as a varying rank decoupling matrix and a change in relative degree. Tools from singularity theory, the theory of foliations, nonsmooth analysis and Lie groups will be developed to address this problem. A key feature will be the use of differential forms instead of vector fields for studying such issues. This should also have payoffs for singular exterior differential systems. Applications to techniques such as the method of intermediate integrals for partial differential equations and equations that change type will be studied from this vantage point. It is worth remarking that both components of the project call for similar mathematics.Successful completion of this project should yield benefits for several current and future technologies. For instance, the fabrication of a quantum computer calls for solutions to precisely the type of problems being studied in the project. Similarly, obtaining a snapshot of chemical reactions at the molecular level also requires the control of quantum processes. Active control of quantum phenomena would also lead to the control of chemical reactions via tuned lasers. This should enhance product specificity in chemical reactions, thereby minimizing the presence of products that are not desired. The control of quantum phenomena is still in its infancy. The project aims to move it along further by linking it concretely to traditional control theory. Conversely, the results of the project should shed light on the control of macroworld systems. This will also have benefits for traditional technologies such as robotics, consumer electronics and electrical networks, since the underlying mathematics is quite similar. From a didactic point of view, the project's contribution will be to show to undergraduates and graduate students alike that advanced mathematics is not an arid field, but is in fact a lively process with palpable connections to real life situations. This should enhance the likelihood of students, especially from under represented groups, to pursue higher education in mathematics, science and technology.