New Stochastic Processes, Partial Differential Equations, and Control Problems Arising in Models of Mechanical Structures Subjected to Vibrations

振动机械结构模型中出现的新随机过程、偏微分方程和控制问题

• 批准号: 0705247
• 负责人: Alain Bensoussan
• 金额: --
• 依托单位: University of Texas at Dallas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-15 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705247&HistoricalAwards=false
• 关键词: New; Stochastic; Processes; Partial; Differential

This project concerns the study of randomly excited nonlinear dynamical systems with memory. Such systems represent viable models for predicting the response of mechanical structures when stressed beyond the elastic limit. A new mathematical approach is considered: stochastic variational inequalities (SVI). The stochastic process solutions of the underlying SVI are essentially continuous diffusion processes with degeneracies and state constraints, for which governing equations can be formulated to describe the evolution in time of their probability distributions. Dissipativity of the system can be used to show existence of the corresponding invariant measure. The uniqueness of this invariant measure and the ergodic property are conjectured to be true for classes of hysteretic systems appearing in applications. Based on the availability of the probability distribution as the solution of the SVI, many practical issues concerning reliability of structures modeled by hysteretic systems can be answered in a systematic fashion. Control problems for SVI are also considered, such as finding the lowest energy input excitation (the so-called critical excitation) that drives the system between prescribed initial and final states within a given time span. Obtaining necessary conditions is a non trivial problem in view of lack of differentiability of the state equations, i.e., optimality conditions are needed for non-smooth dynamical systems. The use of approximate penalty techniques (to remove the constraints) leads to the consideration of degenerate diffusion processes on infinite domains, with challenging questions concerning ergodicity of the corresponding process.The nonlinear behavior of mechanical structures subjected to vibrations is a major concern in designing buildings or plants that must resist earthquakes, ocean waves, the wind, and other random excitations. Many attempts have been made in the past decades, with some successes, in defining universally acceptable safety standards corresponding to potentially occurring environmental loadings. However, the analysis of the underlying mathematics of models involved in these problems is far from complete. The accumulated fatigue that explains the ruin of structures is a nonlinear phenomenon with memory. By providing a thorough understanding of stochastic processes related to the responses of nonlinear systems described by stochastic variational inequalities, this research will establish a solid theoretical foundation for the study of system reliability.

本计画系关于随机激励之记忆非线性动力系统之研究。 这样的系统代表可行的模型，用于预测机械结构的响应时，应力超过弹性极限。 提出了一种新的数学方法：随机变分不等式（SVI）。 潜在的SVI的随机过程的解决方案，本质上是连续的扩散过程退化和状态约束，其中的控制方程可以制定描述其概率分布的时间演变。 系统的耗散性可以用来证明相应的不变测度的存在性。 证明了该不变测度的唯一性和遍历性对于应用中出现的一类滞回系统是成立的。 基于概率分布作为SVI的解的可用性，可以以系统的方式回答关于由滞后系统建模的结构的可靠性的许多实际问题。 也被认为是SVI的控制问题，如寻找最低的能量输入激励（所谓的临界激励），驱动系统在规定的初始和最终状态之间在给定的时间跨度。 鉴于状态方程缺乏可微性，获得必要条件是一个重要的问题，即，非光滑动力系统需要最优性条件。 使用近似罚技术（去除约束）导致考虑无限域上的退化扩散过程，并对相应过程的遍历性提出挑战性问题。受到振动的机械结构的非线性行为是设计必须抵抗地震、海浪、风和其他随机激励的建筑物或工厂时的主要关注点。 在过去几十年中，在确定与可能发生的环境负荷相对应的普遍可接受的安全标准方面进行了许多尝试，并取得了一些成功。 然而，这些问题所涉及的模型的基本数学分析还远远没有完成。 结构的累积疲劳是一种具有记忆性的非线性现象。 通过对由随机变分不等式描述的非线性系统响应的随机过程的深入理解，为系统可靠性的研究奠定了坚实的理论基础。