Control and Optimization of Semi-Dissipative Systems

运输和混合在大气和海洋的循环、环境污染物的扩散、建筑物的通风、化学物质的混合以及自然和工程系统中的许多其他现象中起着核心作用。什么样的速度场能有效地加强或阻止运输和混合，或使数量（如质量浓度或密度分布）达到理想的分布，这些问题引起了越来越多的关注。本项目旨在利用最优控制理论、非线性偏微分方程、非凸非光滑优化等现代技术来解决这些问题。该项目将促进不同研究小组之间的跨学科合作和教育努力，并将帮助培养数学、机械工程和化学工程等领域的高级本科生和研究生，以便在最佳运输和混合、流动控制和计算方法方面进行合作研究。本研究的目的是建立一个创新和严格的数学框架，通过主动控制流体流动来实现最佳的输送和混合。该项目侧重于(1)通过控制不可压缩的Navier-Stokes方程来优化输送和混合，内部（分布式）和边界控制设计用于不同的物理应用；(2)研究由输运方程控制的非耗散标量的被动和主动输运行为，并通过二维Boussinesq近似模拟浮力驱动的流动来解决主动输运问题；(3)应用非光滑正则化来提高控制的稀疏性。研究最优控制的存在性、耦合非线性系统的可微性以及最优性条件的推导，有望为非线性控制、非凸优化和半耗散系统的非光滑分析开辟新的方向。它还将有助于开发有效的计算方法。本项目开发的最优控制综合将使研究其他复杂流动中出现的类似现象成为可能，例如导电流体中的传质、可压缩流动中的热交换和海洋学中的双重扩散，这些都是广大科学家和工程师非常感兴趣的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Transport and mixing play central roles in the circulation of the atmosphere and oceans, the spreading of environmental pollutants, the ventilation in buildings, the mixing of chemical substances, and many other phenomena in natural and engineered systems. The questions of what velocity fields effectively enhance or prevent transport and mixing, or steer quantities (such as mass concentration or density distribution) to a desired distribution, have attracted increasing attention. This project aims to use modern techniques from optimal control theory, nonlinear partial differential equations, and nonconvex and nonsmooth optimization to address these questions. The project will foster interdisciplinary collaboration and education efforts among different research groups and will also help train advanced undergraduate and graduate students across fields including mathematics, mechanical engineering, and chemical engineering to conduct collaborative research in optimal transport and mixing, flow control, and computational methods. The objective of this research is to establish an innovative and rigorous mathematical framework for achieving optimal transport and mixing via active control of fluid flows. The project focuses on (1) optimizing transport and mixing via control of the incompressible Navier-Stokes equations, with internal (distributed) and boundary control designs employed for different physical applications; (2) investigating the transport behavior of non-dissipative scalars, both passive and active, governed by the transport equations and addressing active transport through buoyancy-driven flows, modeled by the 2D Boussinesq approximation; and (3) applying non-smooth regularization to improve the sparsity of the controls. Investigation of the existence of an optimal control, differentiability of the coupled nonlinear system, and derivation of optimality conditions promises to launch new directions in nonlinear control, nonconvex optimization, and nonsmooth analysis of semi-dissipative systems. It will also contribute to the development of effective computational methods. The optimal control synthesis developed in this project will enable the study of similar phenomena arising in other complex flows, such as mass transfer in electrically conducting fluids, heat exchange in compressible flows, and double diffusion in oceanography, that are of great interest to a broad range of scientists and engineers.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.