Optimal Control in Coupled Systems with Moving Interfaces

具有移动界面的耦合系统的最优控制

• 批准号: 1312801
• 负责人: Lorena Bociu
• 金额: $ 18万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312801&HistoricalAwards=false
• 关键词: Optimal; Control; Coupled; Systems; Moving

This project addresses the control of turbulence inside fluid flow in the case of free boundary interaction between a viscous fluid and a moving and deforming elastic body. Reducing and controlling turbulence flow is particularly relevant in the design of small-scale unmanned aircrafts and morphing aircraft wings (e.g. improving the flight of a robobee), and is also of great interest in the medical community (for example, blood flow in a stenosed or stented artery). The proposed problem presents new challenges since it treats the case of a moving and deforming elastic body coupled with a viscous fluid, building on the existing literature on control problems in fluid-structure interactions, which is predominantly focused on the assumption of small but rapid oscillations of the solid body, and therefore assumes that the common interface is static. Due to the presence of the free boundary and the fully nonlinear coupled system, the issue of existence and uniqueness of an optimal control will involve strategies from sensitivity and shape differentiability analysis, on top of techniques from control theory of partial differential equations and well-posedness analysis for fluid-structure interactions. Therefore, the project will be of interest to a broad mathematical and engineering audience. Mathematically, the proposed research will lead to: (i) the construction of quasilinear theory arising in Navier-Stokes equations coupled with waves, (ii) the development of the theory of strong shape derivatives for hyperbolic problems with non-smooth Neumann boundary conditions, which is challenging due to the failure of the Lopatinski condition, and (iii) the study of well-posedness analysis for the first linear model of fluid-elasticity interaction that takes into account the common interface and its curvatures, which are critical for a correct physical interpretation of the coupling. The project will launch new research in the field, since the approach and the techniques introduced for the minimization of drag in the fluid-elasticity interaction can be adjusted and used to investigate inverse or control problems in many other free boundary coupled physical systems, and with different types of controls. The principal investigator has partnered with the North Carolina Museum of Natural Sciences in an effort to create a "Math Day at the Museum", in order to promote and present her research to a broad audience, showcase students' research through posters and presentations, and encourage the participation of women and minorities in the study of math. During these events, the principal investigator will give public presentations on her research, develop and set up demonstrations of real world phenomena illustrating applications of partial differential equations and their control. In addition, the principal investigator plans to promote women and minorities in math, by involving several student groups from North Carolina State University, including the Association for Women in Mathematics, Women in Science and Engineering, and the Society of African American Physical and Mathematical Sciences.

在粘性流体与运动和变形的弹性体之间存在自由边界相互作用的情况下，本项目致力于控制流体内部的湍流流动。减少和控制湍流在小型无人机和变形飞机机翼的设计中尤其重要(例如，改进RoboBee的飞行)，也是医学界非常感兴趣的(例如，狭窄或支架动脉中的血液流动)。提出的问题提出了新的挑战，因为它处理了运动和变形的弹性体与粘性流体耦合的情况，建立在现有的流固相互作用控制问题的文献基础上，这些文献主要集中在固体的小但快速振荡的假设上，因此假设公共界面是静态的。由于自由边界和完全非线性耦合系统的存在，最优控制的存在和唯一性问题将涉及到灵敏度和形状可微性分析的策略，以及偏微分方程控制理论和流固耦合适定性分析的方法。因此，该项目将引起广大数学和工程学观众的兴趣。在数学上，拟议的研究将导致：(I)建立与波耦合的Navier-Stokes方程的拟线性理论；(Ii)发展具有非光滑Neumann边界条件的双曲型问题的强形状导数理论，由于Lopatinski条件的失败，这是具有挑战性的；以及(Iii)考虑公共界面及其曲率的第一个线性流体-弹性相互作用模型的适定性分析的研究，这对于正确的物理解释耦合是至关重要的。该项目将在该领域开展新的研究，因为在流体-弹性相互作用中引入的最小化阻力的方法和技术可以调整，并用于研究许多其他自由边界耦合物理系统中的逆问题或控制问题，并具有不同类型的控制。这位首席研究人员与北卡罗来纳州自然科学博物馆合作，努力创建“博物馆数学日”，以向广大观众宣传和展示她的研究，通过海报和演示展示学生的研究，并鼓励女性和少数族裔参与数学研究。在这些活动期间，首席研究员将公开介绍她的研究，开发和建立现实世界现象的演示，说明偏微分方程及其控制的应用。此外，首席研究员计划让北卡罗来纳州立大学的几个学生团体参与进来，促进数学领域的妇女和少数族裔，包括数学妇女协会、科学和工程妇女协会以及非裔美国人物理和数学科学学会。