Controls and their Large Scale Computation for Long Time Dynamics of Navier-Stokes Equations and Burgers' Equation

9626154 Yan研究者研究由Navier-Stokes方程或Burgers方程控制的不可压缩流。对一些特殊问题进行了深入研究。研究了各种类型的控制目标（速度跟踪、阻力、涡量最小化）和控制参数（分布式控制和边界控制）。控制流的长期动力学是本项目的主要课题。研究了在不同情况下全局实时和分段实时最优控制的长期行为。设计了计算控制问题的数值格式，并对其性能进行了分析。分析了连续系统和离散系统的控制效果；后者特别接近实际控制的设计。采用数值格式来评价控制问题的有效性。数值结果不仅加强了理论结果，而且揭示了更多的受控动力学的定量信息。改进现代飞机的设计和控制可以通过机翼、发动机等的优化设计，或者通过主动控制流过机翼和进入发动机的空气来实现。该项目研究主动控制的基本问题。研究人员开发了新的数学和计算工具，为控制空气动力和水动力系统中的流体运动提供了良好的工程洞察力和实用的设计标准。这需要高性能的计算资源，比不受控制的流的典型建模要大一个数量级。直接的应用也可以在潜艇的外形和推进系统的设计中找到。

9626154 Yan The investigator studies incompressible flows governed by the Navier-Stokes equations or Burgers' equation. Some special issues are studied in depth. Various types of control objectives (minimizing velocity tracking, drag, vorticity) and control parameters (distributed control and boundary control) are studied. Long-time dynamics of controlled flows is the main subject of this project. The long-time behavior of global-in-time and piecewise-in-time optimal controls are studied in different settings. Numerical schemes for computing the control problems are designed and their performance is analyzed. Effectiveness of control is analyzed for both continuous and discrete systems; the latter is particularly close to the design of practical control. Numerical schemes are implemented to evaluate the effectiveness of the control problems. The numerical results both reinforce the theoretical results and reveal more quantitative information of the controlled dynamics. Improving the design and control of modern aircraft can be achieved through optimal design of wings, engines, etc, or through the active control of air flowing over wings and into engines. The project studies issues basic to active control. The investigator develops new mathematical and computational tools that provide good engineering insight and practical design criteria for controlling fluid motions in aerodynamic and hydrodynamic systems. This requires high performance computational resources an order of magnitude greater than typical modeling for uncontrolled flows. Direct applications can also be found in the design of submarine body shape and propelling systems.