Optimization of Pseudospectra

在许多应用中都会出现涉及特征值的优化问题。半定规划是具有正半定约束的实对称矩阵空间中的线性优化问题，近年来受到了广泛的关注。这个项目的重点是在更大的方阵空间中的优化问题，而不是一般对称的。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。本项目主要研究伪谱函数的计算、分析和数值优化。第二个目标是促进一般非光滑、非凸优化问题算法的发展。这项工作的应用出现在许多情况下。我们使用的一个很好的例子是处于颤振状态的波音767模型。当飞机飞得太快时，气动（风）力与飞机结构力的相互作用结合在一起，产生了可能是灾难性的不稳定性，就会发生颤振。通常情况下，像这样的飞机不会飞得这么快，但重要的是要知道如果它飞得这么快会发生什么，以及简单的控制器是否能控制飞机。特征值告诉我们长期的稳定性，但伪光谱提供的信息要多得多，它告诉我们短期的稳定性。在控制器参数上对伪谱进行优化，可以使控制器设计简单而有效地避免不稳定性和防止突变。

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.