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，都可以建模为具有随机激励的动力系统。对这些过程的控制要求我们引导个体成分的集体行为走向特定的最终目标。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，而不能指定其精确的最终值。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，拟议的研究旨在发展理论和技术，使这些系统能够在利用最少的资源（通常是能源）的同时，朝着其参数（位置、速度、温度等）的理想终点分布方向发展。设想了广泛的物理系统和应用，因此，将研究一系列具有线性，非线性，平均场等动力学的数学模型。预计这些工具将影响与上述物理系统和应用程序相关的技术。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-概率密度流动的福克-普朗克方程的问题。服从非平凡动力学的质量和资源的最优运输问题也将被考虑；在这些情况下，状态的随机不确定性仍然存在，而随机激励可以忽略不计。此外，还将研究相互作用粒子和平均场势的问题，以及观测噪声和随之而来的性能限制的影响。理论和工具的发展将利用蒙格-坎托洛维奇最优质量输运、薛定谔桥和由所谓的希尔伯特度量引起的几何随机映射的收缩特性之间的界面的最新进展。

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.