Optimal Control in Coupled Systems with Moving Interfaces

本项目研究粘性流体与运动和变形弹性体之间的自由边界相互作用情况下流体流动内部湍流的控制。减少和控制湍流在小型无人驾驶飞机和变形飞机机翼的设计中尤其重要（例如，改善机器蜜蜂的飞行），并且在医学界也引起了极大的兴趣（例如，狭窄或支架动脉中的血流）。所提出的问题提出了新的挑战，因为它处理的是一个运动和变形的弹性体与粘性流体耦合的情况，建立在关于流固相互作用控制问题的现有文献的基础上，这些文献主要集中在假设固体的小而快速振荡，因此假设共同界面是静态的。由于自由边界和完全非线性耦合系统的存在，最优控制的存在性和唯一性问题将涉及灵敏度和形状可微性分析策略，以及偏微分方程控制理论和流固耦合的适定性分析技术。因此，该项目将对广泛的数学和工程受众感兴趣。从数学上讲，拟议的研究将导致：(i)建立了含波的Navier-Stokes方程的拟线性理论；（ii）发展了具有非光滑Neumann边界条件的双曲型问题的强形状导数理论，这一理论由于Lopatinski条件的失败而具有挑战性；（iii）研究了考虑共同界面及其曲率的第一个流弹性相互作用线性模型的适定性分析。这对于对耦合的正确物理解释至关重要。该项目将在该领域开展新的研究，因为所介绍的流体-弹性相互作用中阻力最小化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或控制问题，并具有不同类型的控制。这位首席研究员与北卡罗来纳自然科学博物馆合作，努力创建一个“博物馆数学日”，以便向广大观众推广和展示她的研究，通过海报和演讲展示学生的研究，并鼓励妇女和少数民族参与数学研究。在这些活动中，首席研究员将公开展示她的研究，开发和建立真实世界现象的演示，说明偏微分方程及其控制的应用。此外，首席研究员计划通过与北卡罗莱纳州立大学的几个学生团体合作，包括数学女性协会、科学与工程女性协会和非裔美国人物理与数学科学协会，来促进女性和少数族裔在数学方面的发展。

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.