耗散映射的刻画及其在耗散哈密顿系统扰动中的应用
Characterization of the dissipative mappings and their application to perturbations of dissipative-Hamiltonian systems
摘要：
本文给出了判别矩阵对$X$和$Y$存在$Δ\in \mathbb C^{n,n}$使得$Δ+Δ^*$是半正定的和$ΔX=Y$的充要条件.这样的$Δ$被称为将$X$变为$Y$的耗散映射。给出了所有耗散映射的集合的两种不同的刻画，并利用它们刻画了唯一的具有极小Frobenius范数的耗散映射。最小范数的耗散映射，然后确定的距离渐近不稳定的耗散哈密顿系统在一般的结构保持扰动。我们通过一些数值例子说明了我们的结果，并将它们与Mehl，Mehrmann和Sharma（Stability Radii for Linear哈密顿系统with Dissipation Under Structure-Preserving Perturbations，SIAM J. Mat. Anal.应用程序\ 37（4）：1625-1654，2016）。
英文摘要：
In this paper, we find necessary and sufficient conditions to identify pairs of matrices $X$ and $Y$ for which there exists $Δ\in \mathbb C^{n,n}$ such that $Δ+Δ^*$ is positive semidefinite and $ΔX=Y$. Such a $Δ$ is called a dissipative mapping taking $X$ to $Y$. We also provide two different characterizations for the set of all dissipative mappings, and use them to characterize the unique dissipative mapping with minimal Frobenius norm. The minimal-norm dissipative mapping is then used to determine the distance to asymptotic instability for dissipative-Hamiltonian systems under general structure-preserving perturbations. We illustrate our results over some numerical examples and compare them with those of Mehl, Mehrmann and Sharma (Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations, SIAM J. Mat. Anal. Appl.\ 37 (4): 1625-1654, 2016).
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