概率工程力学中的随机控制方法

Overview: In an extremely wide range of applications, mechanical systems are fundamentally affected by vibrations. For instance, components (such as pipes) in a power plant must be designed to be structurally robust under seismic vibrations. Vibrations also cause mechanical systems to accumulate fatigue and eventually fail. The importance of these issues has motivated a lot of effort in modeling. From the modeling point of view, keeping track of the impact of past vibrations requires a specific description of the mechanical systems under consideration. The state of the system must be described by a randomly forced dynamical system with memory. Random forcing here expresses the stochastic nature of the vibrations that apply to the mechanical structures. In this framework, the difficulty is to handle dynamical systems with memory. A huge engineering literature has been devoted to this topic. The project mainly focuses on an important model, referred to the elastic-plastic oscillator, which appears in various applications. Some members of the team already started to cooperate on the so-called elastic-perfectly-plastic oscillator. In this context, they identified stochastic variational inequalities as the right mathematical tool to deal with the model. Our objective in this proposal is to extend the research to the analysis of a new class of Hamilton-Jacobi-Bellman Equations and Free Boundary Problems related to stochastic control methods in earthquake engineering. Intellectual merits: The proposed work will provide a solid mathematical framework regarding stochastic control methods for non-smooth stochastic systems including elasto-perfectly-plastic and bilinear elasto-plastic oscillators. Within this rigorous framework, we expect that we will be able to assess quantities of interest related to the risk analysis of failure of mechanical structures. Broader Impact: With our research activities, we propose an outreach program designed to communicate to researchers from applied mathematics and probabilistic engineering mechanics communities (via communications in publications and conferences) to a more general audience of students (via a course on Stochastic Control Methods for Random Mechanics). Also, we will release publicly the results and prototypes Softwares. While our work is driven by specific mechanical problems, we believe that our framework for stochastic control of these non-smooth dynamical systems is more broadly applicable. Applications that could make use of this framework abound in many areas of science and engineering from chemical systems to biological systems to fluid dynamics to finance.

在很多实际应用的工程问题中,力学系统从根本上的受到振动的影响。例如发电厂的线路和传输管道必须设计使得其在地震波的影响下仍然是非常稳定和坚固的。振动的影响使得力学系统的疲劳累积并最终可能使得其奔溃。对这些系统的描述,必须利用带有记忆效果并在随机外力作用下的动力系统来描述。这里随机力用来表达作用在这个系统上的振动是具有随机性质的。而在这个框架下,最难的是如何研究和处理带有记忆的动力系统。 这个项目主要是针对其中一个重要的模型来展开的,这个模型是实际应用中经常出现的弹塑性振子模型。 我们的研究小组中的一些成员已经就理想的弹塑性振子模型展开了研究，他们利用随机变分不等式作为数学工具来处处理这个模型。而我们现在这个项目的将要推广上述的研究，并将分析和研究模型得到的一类新的HJB方程， 并研究一些和该模型中出现的和随机控制有关的自由边值问题，我们这里研究的问题和地震工程密切相关。

在很多实际应用的工程问题中,力学系统从根本上的受到振动的影响。例如发电厂的线路和传输管道必须设计使得其在地震波的影响下仍然是非常稳定和坚固的。振动的影响使得力学系统的疲劳累积并最终可能使得其奔溃。对这些系统的描述,必须利用带有记忆效果并在随机外力作用下的动力系统来描述。这里随机力用来表达作用在这个系统上的振动是具有随机性质的。而在这个框架下,最难的是如何研究和处理带有记忆的动力系统。 这个项目主要是针对其中一个重要的模型来展开的,这个模型是实际应用中经常出现的弹塑性振子模型。 我们的研究小组中的一些成员已经就理想的弹塑性振子模型展开了研究，他们利用随机变分不等式作为数学工具来处处理这个模型。而我们现在这个项目的将要推广上述的研究，并将分析和研究模型得到的一类新的HJB方程， 并研究一些和该模型中出现的和随机控制有关的自由边值问题，我们这里研究的问题和地震工程密切相关。