Control and Optimization of Semi-Dissipative Systems

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

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.