Optimal Control Problems with Linear Bellman Equations

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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.