CAREER: Control and Sensitivity Analysis for Fluid-Elasticity Interactions and Fluid-Solid Mixtures

许多重要的生物学和物理问题涉及到流体与弹性体的相互作用，或流体在可变形的多孔介质中的流动。这些问题分别称为流固相互作用问题和流固混合问题。在大多数这些应用中，最终目标是对所考虑的过程进行最优化或最优控制，以及相关的敏感性分析(关于相关的物理/生物参数)调查。例如，设计小型无人驾驶飞机和变形飞机机翼，如机器蜜蜂，将狭窄或支架动脉内的血流湍流降至最低，或优化血液流动的压力，并研究相关生物参数对眼睛筛板的影响，这些因素被认为与青光眼的发展有关。这项研究计划解决移动边界流体-弹性相互作用和流体通过生物组织流动的敏感性分析、控制和优化相关问题。该项目将专注于眼部筛板生物力学和血流动力学之间的耦合的具体例子，其中对重要参数(如血流和眼内压和板层后压)进行敏感性分析将进一步了解青光眼的病因和进展，并使预防或治疗青光眼的新方法成为可能。这项研究也将适用于其他生物流体-固体混合物，如软骨、骨骼和工程组织。动边界流体-弹性相互作用和多孔介质渗流是具有Banach空间参数的非线性动力系统，其灵敏度分析和控制的大部分问题在该领域都是公开的。该项目将(I)通过建立灵敏度分析和最优性条件的理论框架来推进这些非线性系统的最优控制问题的理论；(Ii)对筛板模型进行灵敏度分析以确定重要的生物学参数，并利用所获得的结果来开发和解决青光眼发展中可能的控制策略。最优解的适定性，代价泛函和状态关于分布控制和边界控制的Frechet可微性，以及伴随系统的推导和适定分析将被讨论。这些结果将为所有灵敏度计算、参数估计和基于导数的优化算法提供起点。

Many important biological and physical problems involve either the interaction of a fluid with an elastic body, or the flow of a fluid through a deformable, porous medium. These problems are known as fluid-structure interactions and fluid-solid mixture problems, respectively. In most of these applications, the ultimate goal is the optimization or optimal control of the considered process, as well as related sensitivity analysis (with respect to relevant physical/biological parameters) investigations. Examples include the design of small-scale unmanned aircrafts and morphing aircraft wings, like robo-bees, minimizing turbulence in blood flow within a stenosed or stented artery, or optimizing the pressure of the blood flow and investigating the influence of pertinent biological parameters in the case of the lamina cribrosa in the eye, where these factors are believed to be related to the development of glaucoma. This research program addresses sensitivity analysis, control, and optimization-related problems for moving boundary fluid-elasticity interactions and fluid flows through biological tissues. The project will focus on the particular example of the coupling between biomechanics and hemodynamics in the lamina cribrosa in the eye, where performing sensitivity analysis with respect to important parameters (like blood flow and intraocular and retrolaminar pressures) will further the understanding of the cause and progression of glaucoma, and enable novel means for preventing or treating glaucoma. This research will also be applicable to other biological fluid-solid mixtures such as cartilages, bones, and engineered tissues. Moving boundary fluid-elasticity interactions and porous media flows are nonlinear dynamical systems with Banach space parameters, and most questions related to their sensitivity analysis and control are open in the field. This project will (i) advance the theory of optimal control problems subject to these nonlinear systems by developing the theoretical framework for sensitivity analysis and optimality conditions; (ii) perform sensitivity analysis on the lamina cribrosa model to identify the important biological parameters, and use the obtained results to develop and address possible control strategies in the development of glaucoma. Well-posedness of optimal solutions, Frechet differentiability of cost functionals and state with respect to both distributed and boundary controls, and derivation and well-posedness analysis of adjoint systems will be addressed. These results will provide the starting point for all sensitivity calculations, parameter estimation, and derivative-based optimization algorithms.