通过黎曼optimization的最近$Ω$ -稳定矩阵
Nearest $Ω$-stable matrix via Riemannian optimization
摘要：
研究了寻找与某个矩阵$A$最近的$Ω$ -稳定矩阵的问题，即，所有特征值都在规定的闭集$Ω$中的最近矩阵。距离是用Frobenius范数来衡量的。一个重要的特殊情况是找到最近的Hurwitz或Schur稳定矩阵，这在系统理论中有应用。我们描述了一个重新制定的任务作为一个最优化问题的正交（或酉）矩阵的黎曼流形。然后，可以使用来自黎曼优化理论的标准方法来解决该问题。由此产生的算法是非常快的小规模和中等规模的矩阵，并直接返回一个舒尔分解的最小化，避开了数值困难与高的多重性特征值。
英文摘要：
We study the problem of finding the nearest $Ω$-stable matrix to a certain matrix $A$, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set $Ω$. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.
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