Variational Analysis, Optimization of Eigenvalues, and Robust Stability

提出的研究的主要焦点是分析和优化各种措施的鲁棒稳定性的线性系统，即线性动力系统，随着时间的推移。这里的稳定性一般是指系统在接受脉冲后恢复到稳定状态的能力，例如，结构在地震后停止振动的能力。术语“鲁棒稳定性”指的是试图平衡系统的暂态稳定性和渐近稳定性的稳定性概念。线性系统的渐近稳定性与底层系统的特征值有很好的关系。然而，在线性动力学中，人们早就认识到，当相关的线性变换高度非正态时，特征值很少或根本没有给出系统瞬态行为的信息。从工程设计和控制的角度来看，这种暂态行为是不可接受的。这引起了现代对瞬态稳定性的关注，它通常指的是有界瞬态峰值和/或有界瞬态增长/衰减率等性质。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者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.