联合谱半径的平方和近似
Approximation of the joint spectral radius using sum of squares
摘要：
我们提供了一个渐近紧，计算效率高的近似联合谱半径的一组矩阵使用平方和（SOS）编程。该方法是基于搜索SOS多项式，证明同时收缩的矩阵的有限集。我们提供了一个统一的几个较早的结果，是独立的矩阵的数量的近似的质量上的界限。此外，我们提出了我们的近似方案和早期的技术，包括使用常见的二次李雅普诺夫函数和基于矩阵提升的方法之间的比较。理论结果和数值研究表明，我们的方法产生更紧密的近似。
英文摘要：
We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices. Additionally, we present a comparison between our approximation scheme and earlier techniques, including the use of common quadratic Lyapunov functions and a method based on matrix liftings. Theoretical results and numerical investigations show that our approach yields tighter approximations.
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