US-Turkey Cooperative Research: Semi-Analytical Treatments of Nonlinear Stochastic Dynamical Systems with Application to Control Problems

0217453说明：该奖项是为了支持特拉华州纽瓦克特拉华大学机械工程系孙建乔博士和土耳其伊斯坦布尔技术大学工程科学系Ghazanfer Unal博士共同开展的一项研究项目。本项目的目标是发展分析和数值方法来研究随机动力系统。研究人员计划将李氏的确定性微分方程理论扩展到非线性随机动力系统。具体地说，他们将以一种与随机系统的范式理论保持一致的方式，将他们之前关于近似对称性的结果推广到随机动力系统。随机动力系统的近似对称性的一个直接用途是获得它们的近似守恒量。此外，他们将结果推广到福克-普朗克方程，以获得解析近似解。Unal教授在项目中得到的解析解将与孙博士研究的细胞作图方法相结合。近似对称性和相关的守恒定律可以帮助确定可以应用单元映射方法的子空间。这有望大大减少使用细胞映射方法分析和控制非线性随机系统的计算工作量。范围：成功地确定近似对称性和相关的守恒定律将使确定一个子空间成为可能，在这个子空间中，细胞映射可以用比目前可能的少得多的资源来应用。两位科学家具有互补的优势，因为孙博士的专长是细胞制图，而Unal博士的专长是李氏理论以及动力系统的对称性和近似对称性。研究成果将与两所大学的研究生课程相结合，并可能在两所大学之间开展研究生交流项目。Unal博士将在第一年和第三年前往美国，孙教授将在第二年前往土耳其。该项目将导致他们之间的长期合作，因为他们将解决需要新见解和先进计算工具的难题。

0217453SunDescription: This award is for support of a joint research project by Dr. Jian-Qiao Sun, Department of Mechanical Engineering at the University of Delaware, Newark, Delaware and Dr. Ghazanfer Unal, Engineering Science Department at Istanbul Technical University, Istanbul, Turkey. The goal of this project is to develop analytical and numerical methods to study the stochastic dynamical systems. The investigators plan to extend Lie's theory for deterministic differential equations to nonlinear stochastic dynamical systems. Specifically, they will extend their previous results on the approximate symmetries to stochastic dynamical systems in a manner which will remain consistent with the normal form theory of stochastic systems. An immediate use of the approximate symmetries of the stochastic dynamical systems would be to obtain their approximate conserved quantities. Furthermore, they will extend the results to the Fokker-Planck equation in order to obtain analytical approximate solutions. The analytical solutions to be obtained in the project by Professor Unal will be integrated with the cell mapping method studied by Dr. Sun. The approximate symmetries and the related conservation laws can help identify a sub-space in which the cell mapping method can be applied. This is expected to substantially reduce the computational effort for analysis and control of nonlinear stochastic systems by using the cell mapping method.Scope:Success in determining approximate symmetries and the related conservation laws would make it possible to identify a subspace in which cell mapping can be applied with much less resources than currently possible. The two scientists have complementary strengths, as Dr. Sun's expertise is in cell mapping, while Dr. Unal's expertise is in Lie's theory and in symmetries and approximate symmetries of dynamical systems. Research results will be integrated with graduate programs at both universities, and an exchange program of graduate students between the two universities may also result. Dr. Unal will travel to the United States in the first and third years while Prof. Sun will travel to Turkey in the second year. The project will lead to long-term collaboration between them since they will tackle difficult problems that require new insights, as well as advanced computational tools.