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

本课题研究的是随机激励非线性动力系统的记忆问题。这种系统为预测机械结构在应力超过弹性极限时的响应提供了可行的模型。提出了一种新的数学方法：随机变分不等式（SVI）。底层SVI的随机过程解本质上是具有退化和状态约束的连续扩散过程，其控制方程可以描述其概率分布随时间的演变。系统的耗散率可用来表示相应不变测度的存在性。对于应用中出现的滞回系统，我们推测该不变测度的唯一性和遍历性是成立的。基于概率分布作为SVI解的有效性，许多有关滞回系统建模结构可靠度的实际问题可以系统地回答。还考虑了SVI的控制问题，例如寻找在给定时间范围内驱动系统在规定的初始状态和最终状态之间的最低能量输入激励（所谓的临界激励）。由于状态方程缺乏可微性，求必要条件是一个非平凡问题，即非光滑动力系统需要最优性条件。使用近似惩罚技术（去除约束）导致考虑无限域上的退化扩散过程，并提出了有关相应过程遍历性的挑战性问题。机械结构在振动下的非线性行为是设计建筑物或工厂时必须考虑的一个主要问题，这些建筑物或工厂必须抵抗地震、海浪、风和其他随机激励。在过去的几十年里，已经进行了许多尝试，并取得了一些成功，以确定与可能发生的环境负荷相对应的普遍接受的安全标准。然而，对涉及这些问题的模型的基础数学的分析还远远没有完成。累积疲劳解释了结构破坏是一种具有记忆性的非线性现象。通过对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，本研究将为系统可靠性的研究奠定坚实的理论基础。

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.