Optimization of Pseudospectra

涉及特征值的优化问题在许多应用中出现。半定规划是真实的对称矩阵空间中具有半定正约束的线性优化问题，近年来受到人们的广泛关注。 这个项目集中在较大的方阵空间中的优化问题，一般不对称。在这种情况下，众所周知，本征值，它提供了有关动力系统的渐近行为的信息，是不作为pseudospectrum，它提供了一个更鲁棒的测量系统的行为有用。本项目的重点是伪谱函数的计算、分析，特别是数值优化。第二个目标是推进一般非光滑、非凸优化问题算法的发展，这项工作的应用出现在许多场合。 一个很好的例子是一架处于颤振状态的波音767飞机模型，当飞机飞得太快时，空气动力（风）与飞机结构力的相互作用联合收割机会产生一种不稳定性，这种不稳定性会造成灾难性的后果。 正常情况下，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，以及一个简单的控制器是否可以控制飞机。 本征值告诉我们长期的稳定性，但伪谱的信息量更大，告诉我们短期的稳定性。 优化控制器参数的伪谱可以导致控制器的设计，该控制器在避免不稳定性和防止灾难方面是简单而有效的。

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.