Optimal Control Problems with Linear Bellman Equations

该提案的目的是通过开发一种创新方法（称为维数分解法）来解决随机动态系统建模和仿真中的一般随机特征值问题，从而在爱荷华州大学的计算力学领域实现核心竞争力。 所提出的努力将基于：（1）新的分解方法的低维近似的一般复值特征值的随机特征值问题的特征解;（2）新的多点分解和单项预条件的概率特征的特征解;和（3）新的设计灵敏度公式的随机特征解的概率测度的分析梯度。 拟议的研究是雄心勃勃的和新颖的，在根本上不同于这一领域的大多数先前的研究。 要开发的方法将解决高度非线性的输入-输出转换、无限数量的随机变量或字段以及随机输入的任意大的不确定性。 由于创新配方的分析推导出的随机设计灵敏度，随后的动态系统的优化可以进行采用任何标准的基于梯度的算法。 该分解方法将有助于解决工程和科学中的大规模、多学科、随机特征值问题，对民用、汽车和航空航天基础设施等众多商业和工业应用具有重要意义。 潜在的工程应用包括民用结构的分析和设计;地面车辆系统的噪声-振动-粗糙度;航空航天结构的疲劳耐久性;以及微电子和微机电系统的可靠性。 除了工程学之外，潜在的应用还包括核物理、数论、计算生物学和计算金融等。 因此，这里提出的研究将对一些具有国家意义的领域产生积极影响。 该项目创造的知识的转移和传播将通过与行业的持续合作，在ASME会议上组织研讨会，期刊出版物，在主要会议和机构的演讲和出版物以及学生教育来进行。 与两个政府和工业实验室的伙伴关系将使本项目中开发的基本方法得以实施，以解决几个大规模的工业问题。 教育目标包括招收博士生。来自代表性不足的少数民族或妇女群体的学生，在爱荷华州大学主要工程项目的升级课程中实施该项目的软件工具，并撰写一本研究专著。

The objective of the proposal is to achieve a core competency in the field of computational mechanics at The University of Iowa through the development of an innovative method, referred to as the dimensional decomposition method, for solving a general random eigenvalue problem in modeling and simulation of stochastic dynamic systems. The proposed effort will be based on: (1) new decomposition method for lower-dimensional approximations of general complex-valued eigensolutions of random eigenvalue problems; (2) new multipoint decomposition and monomial preconditioner for probabilistic characteristics of eigensolutions; and (3) new design sensitivity formulation for analytic gradients of probabilistic measures of random eigensolutions. The proposed research is ambitious and novel, differing in fundamental ways from most prior research in this area. The methods to be developed will address highly nonlinear input-output transformations, an unlimited number of random variables or fields, and arbitrarily large uncertainty of random input. Due to innovative formulation of the analytically derived stochastic design sensitivities, subsequent optimization of dynamic systems can be conducted employing any standard gradient-based algorithm. The decomposition method will aid in solving large-scale, multidisciplinary, stochastic eigenvalue problems in engineering and science.The proposed research will be of significant benefit to numerous commercial and industrial applications, such as civil, automotive, and aerospace infrastructure. Potential engineering applications include analysis and design of civil structures; noise-vibration-harshness of ground vehicle systems; fatigue durability of aerospace structures; and reliability of microelectronics and micro-electro-mechanical systems. Beyond engineering, potential applications include nuclear physics, number theory, computational biology, and computational finance, among others. Therefore, the research proposed here will positively impact a number of areas of national significance. The transfer and dissemination of knowledge created by this project will take place through continued collaboration with industries, organization of symposia in ASME conferences, journal publications, presentations and publications at major conferences and institutions, and student education. Partnerships with two government and industrial laboratories will enable implementation of the basic methods developed in this project to resolve several large-scale industrial problems. The educational goals comprise recruitment of a Ph. D. student from underrepresented minority or women groups, implementation of software tools from this project in upgrading courses in The University of Iowa's principal engineering programs, and authoring a research monograph.