Variational Analysis, Optimization of Eigenvalues, and Robust Stability

所提出的研究的主要重点是分析和优化的各种措施的鲁棒稳定性的线性系统，演变的时间，即，线性动力系统 这里的稳定性通常是指系统在接收到脉冲之后返回到稳定状态的能力，例如，地震后建筑物停止振动的能力。 术语“鲁棒稳定性”是指试图平衡系统的瞬时稳定性与渐近稳定性的稳定性概念。 线性系统的渐近稳定性与其特征值之间有着良好的关系。 然而，在线性动力学中，当相关的线性变换是高度非正态的时，特征值给出很少或没有关于系统的瞬态行为的信息。 从工程设计和控制的角度来看，这种瞬态行为是完全不可接受的。 这引起了对暂态稳定性的现代关注，暂态稳定性通常指的是有界暂态峰值和/或有界暂态增长/衰减率等特性。 PI和他的合作者已经为这项研究的进一步进展建立了丰富的技术，结果和数值工具基础。 他们建议继续这项工作，并为工程设计和控制从业人员提供有效的数值工具。

The main focus of the proposed research is the analysis and optimization of various measures of robust stability in linear systems that evolve in time, i.e., linear dynamical systems. Here stability generally refers to the ability of a system to return to steady state after receiving an impulse, e.g., the ability of a structure to stop vibrating after an earthquake. The term "robust stability" refers to notions of stability that attempt to balance the transient stability of a system with its asymptotic stability. The asymptotic stability of a linear system has a well-established relationship to the eigenvalues of the underlying system. However, it has long been recognized in linear dynamics that when the associated linear transformation is highly non-normal the eigenvalues give little or no information about the transient behavior of the system. This transient behavior can be quite unacceptable from the perspective of engineering design and control. This has given rise to the modern focus on transient stability, which generally refers to such properties as bounded transient peaks and/or bounded transient growth/decay rates. The PI and his collaborators have already established a rich foundation of techniques, results, and numerical tools for further progress in this research. They propose to continue this work and to provide efficient numerical tools for engineering design and control practitioners.