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

许多重要的生物学和物理学问题涉及流体与弹性体的相互作用，或流体通过可变形多孔介质的流动。 这些问题分别称为流固耦合问题和流固混合问题。 在大多数这些应用中，最终目标是所考虑的过程的优化或最佳控制，以及相关的灵敏度分析（相对于相关的物理/生物参数）调查。例子包括设计小型无人驾驶飞机和变形飞机机翼，如机器蜜蜂，最大限度地减少狭窄或支架动脉内血流的湍流，或优化血流的压力，并研究眼睛中筛板的相关生物参数的影响，其中这些因素被认为与青光眼的发展有关。 该研究项目针对移动边界流体弹性相互作用和通过生物组织的流体流动的灵敏度分析，控制和优化相关问题。 该项目将重点关注眼睛筛板中生物力学和血液动力学之间耦合的特定示例，其中对重要参数（如血流量和眼内压和椎板后压）进行敏感性分析将进一步了解青光眼的原因和进展，并实现预防或治疗青光眼的新方法。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流动是一类具有Banach空间参数的非线性动力系统，其灵敏度分析和控制问题是该领域的一个重要问题。本项目将（i）通过开发灵敏度分析和最优性条件的理论框架，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，以识别重要的生物参数，并使用所获得的结果开发和解决青光眼发展中可能的控制策略。适定性的最优解，Frechet可微的成本泛函和状态相对于分布式和边界控制，推导和适定性分析的伴随系统将得到解决。 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.