有限维时变线性系的包围控制
Ensemble Control of Finite Dimensional Time-Varying Linear Systems
摘要：
在这篇文章中，我们研究了同时操纵不可数族有限维时变线性系的问题。我们称这类控制问题为包围控制，这是一个来自核磁共振（NMR）光谱和成像（MRI）中自旋动力学研究的概念。本课题涉及到用相同的开环控制输入控制参数化动力系统的连续体。从数学控制理论的观点来看，这类问题是具有挑战性的，因为它需要通过使用相同的控制函数在无限维状态空间中的感兴趣的点之间操纵连续的动力系统。这种控制的存在提出了系综可控性的基本问题。本文给出了时变线性系系综控制的充要可控条件和相应的解析最优控制律。我们表明，系综能控性与表征系统动力学的算子的奇异值有关。此外，我们研究了谐振子的最优系综控制问题，以证明我们的主要结果。我们表明，最佳的解决方案是有关的时间-频率有限的信号和扁长球体波函数的研究。系统地研究系综控制系统可以直接应用于具有参数不确定性的系统以及在相干光谱学和量子信息处理中出现的量子控制系统的广泛领域。这些问题中出现的新的数学结构是控制理论新发展的极好动力。
英文摘要：
In this article, we investigate the problem of simultaneously steering an uncountable family of finite dimensional time-varying linear systems. We call this class of control problems Ensemble Control, a notion coming from the study of spin dynamics in Nuclear Magnetic Resonance (NMR) spectroscopy and imaging (MRI). This subject involves controlling a continuum of parameterized dynamical systems with the same open-loop control input. From a viewpoint of mathematical control theory, this class of problems is challenging because it requires steering a continuum of dynamical systems between points of interest in an infinite dimensional state space by use of the same control function. The existence of such a control raises fundamental questions of ensemble controllability. We derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for ensemble control of time-varying linear systems. We show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics. In addition, we study the problem of optimal ensemble control of harmonic oscillators to demonstrate our main results. We show that the optimal solutions are pertinent to the study of time-frequency limited signals and prolate spheroidal wave functions. A systematic study of ensemble control systems has immediate applications to systems with parameter uncertainties as well as to broad areas of quantum control systems as arising in coherent spectroscopy and quantum information processing. The new mathematical structures appearing in such problems are an excellent motivation for new developments in control theory.
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