Polynomial control on stability, inversion and powers of matrices on simple graphs
Polynomial control on stability, inversion and powers of matrices on simple graphs
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DOI：
10.1016/j.jfa.2018.09.014
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发表时间：
2017-05
影响因子：
1.7
通讯作者：
C. Shin;Qiyu Sun
中科院分区：
文献类型：
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作者：
C. Shin;Qiyu Sun
大规模的空间分布网络出现在各种科学和工程问题中，如无线传感器网络和智能电网。它们的大部分特征可以用它们的状态空间矩阵的性质来描述，这些矩阵的元素在图的顶点集中有索引。本文引入了一类新的Beurling型代数，它们包含多项式非对角衰减的连通简单图上的矩阵，并证明了它们是B（p），1≤ p≤∞的Banach子代数，B（p）是所有p-可和序列空间上的所有有界算子空间.状态空间矩阵的渐近稳定性是空间分布网络鲁棒性的基本假设。本文建立了Beurling代数中矩阵在不同指数1≤ p≤∞下的稳定性之间的等价性，并对稳定性下界进行了定量分析.范数控制反演的引入在工程实践中起着至关重要的作用。本文证明了B（B ~ 2）的Beurling子代数中的矩阵具有模控逆，并找到了一个模控多项式，其次数接近最优.矩阵幂的多项式估计对于空间分布网络的数值实现具有重要意义。本文将我们的结果应用于模控逆，得到了Beurling代数中矩阵幂的多项式估计。多项式估计是关于复变函数卷积幂的非交换扩展，适用于空间分布网络上的平稳马尔可夫链中从一个智能体跳到另一个智能体的概率估计.
Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B (ℓ p), 1≤ p≤∞, the space of all bounded operators on the space ℓ p of all p-summable sequences. The ℓ p-stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among ℓ p-stabilities of matrices in Beurling algebras for different exponents 1≤ p≤∞, with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B (ℓ 2) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network.