Optimization of Pseudospectra

涉及特征值的优化问题在许多应用中都会出现。半定规划是实对称矩阵空间中的线性优化问题，具有半正定约束，近年来，人们的注意力集中在半定规划上。这个项目集中在方阵的较大空间中的优化问题，而不是一般的对称。在这种情况下，众所周知，提供关于动力系统渐近行为的信息的特征值不如伪谱有用，伪谱提供了系统行为的更稳健的度量。本项目的重点是伪谱函数的计算、分析，特别是数值优化。第二个目标是促进一般非光滑、非凸优化问题的算法的发展。这项工作在许多情况下都有应用。我们使用的一个很好的例子是波音767飞机的颤振状态。当飞机飞行得如此之快，以至于空气动力(风)力与飞机内的结构力的相互作用结合在一起产生不稳定时，就会发生颤振，这可能是灾难性的。通常情况下，像这样的飞机不会飞得这么快，但知道如果它飞了会发生什么，以及一个简单的控制器是否能控制飞机，这一点很重要。本征值告诉我们长期稳定性，但伪谱提供的信息要多得多，它告诉我们短期稳定性。通过对控制器参数进行伪谱优化，可以设计出一种既简单又能有效避免失稳和防止灾难的控制器。

Optimization problems involving eigenvalues arise in manyapplications. In recent years, attention has focused onsemidefinite programs, which are linear optimization problems inthe space of real symmetric matrices, with positive semidefiniteconstraints. This project focuses on optimization problems in thelarger space of square matrices, not generally symmetric. In thiscontext, it is well known that eigenvalues, which provideinformation about asymptotic behavior of dynamical systems, arenot as useful as pseudospectra, which provide a more robustmeasure of system behavior. This project focuses on thecomputation, analysis and, especially, numerical optimization ofpseudospectral functions. A secondary goal is to advance thedevelopment of algorithms for general nonsmooth, nonconvexoptimization problems.Applications of this work arise in many contexts. A good examplethat we have used is a model of a Boeing 767 at a flutter condition.Flutter occurs when the plane flies so fast that the interactionof the aerodynamic (wind) forces with the structural forces inthe airplane combine to generate an instability which could becatastrophic. Normally, a plane like this does not fly so fast butit is important to know what would happen if it did, and whether asimple controller could bring the plane under control. Eigenvalues tell us about stability in the long term, but pseudospectra are far more informative, telling us about stability in the short term. Optimizing the pseudospectra over the parameters of the controller may result in the design of a controller that is simple yet effective in avoiding the instability and preventing catastrophe.