Theory and Techniques for Controlling the Collective Behavior of Dynamical Systems under Stochastic Uncertainty

物理系统，如经历热噪声的微机械部件、粒子束和激光驱动的分子反应都可以被建模为具有随机激励的动力学系统。要控制这些过程，就需要我们引导个体成分的集体行为朝着特定的最终目标前进。由于其初始状态的随机不确定性以及各种随机效应（测量噪声、随机强迫、波动耗散现象等），人们只能希望能够指定相应参数的目标分布，而不是它们的精确最终值。此外，对于这种固有的随机系统，通常情况下，控制策略的性能只能在统计意义上量化。因此，拟议的研究旨在开发理论和技术，将允许转向这样的系统对他们的参数（位置，速度，温度等）所需的端点分布。同时利用最少量的资源，通常是能量。一个广泛的物理系统和应用的设想，因此，一系列的数学模型与线性，非线性，平均场等动态将被研究。该研究的一个具体技术目标是研究随机系统的可控性和最优控制，即能够将其状态从给定的初始概率密度引导到最终目标密度。实际上，这是控制指定状态概率密度流的福克-普朗克方程的问题。也将考虑服从非平凡动力学的质量和资源的最优运输问题;在这些状态的随机不确定性仍然存在，而随机激励变得可以忽略不计。此外，还将研究相互作用粒子和平均场势的问题，以及观测噪声的影响和随之而来的性能限制。理论和工具的发展将利用最近的进展之间的接口Monge-Kantorovich最佳质量传输，薛定谔桥，和收缩性质的随机映射的几何形状引起的所谓的希尔伯特度量。

Physical systems such as micro-mechanical components experiencing thermal noise, particle beams, and laser-driven molecular reactions can all be modeled as dynamical system with stochastic excitation. The control of such processes requires that we steer the collective behavior of the individual constituents towards a specified terminal goal. Due to the stochastic uncertainty of their initial state as well as due to various stochastic effects (measurement noise, stochastic forcing, fluctuation-dissipation phenomena, etc.) one can only hope to be able to specify a target distribution for corresponding parameters and not their precise final values. Further, for such inherently stochastic systems, it is often the case that the performance of a control strategy can only be quantified in a statistical sense. Thus, the proposed research aims at developing theory and techniques that will allow steering such systems towards a desired end-point distribution for their parameters (position, velocity, temperature, etc.) while utilizing minimal amount of resources, typically energy. A wide range of physical systems and applications are envisioned and, accordingly, a range of mathematical models with linear, nonlinear, mean-field, etc. dynamics will be studied. The tools are expected to impact technology that relates to the aforementioned physical systems and applications.A specific technical goal of the proposed research is to study the controllability and the optimal control of stochastic systems in the sense of being able to steer their state from a given initial probability density to a final target one. In effect, this is the problem to control the Fokker-Planck equation that specifies the flow of the state-probability density. Problems of optimal transport for mass and resources that obey nontrivial dynamics will also be considered; in these the stochastic uncertainty for the state remains while the stochastic excitation becomes negligible. Furthermore, problems of interacting particles and mean-field potentials will also be studied as will be the effect of observation noise and ensuing limits to performance. The development of theory and tools will make use of recent advances at the interface between Monge-Kantorovich optimal mass transport, Schroedinger bridges, and the contractive properties of stochastic maps in the geometry induced by the so-called Hilbert metric.