We develop an eigenvalue-based approach for the stability assessment and stabilization of linear systems with multiple delays and periodic coefficient matrices. Delays and period are assumed commensurate numbers, such that the Floquet multipliers can be characterized as eigenvalues of the monodromy operator and by the solutions of a finite-dimensional non-linear eigenvalue problem, where the evaluation of the characteristic matrix involves solving an initial value problem. We demonstrate that such a dual interpretation can be exploited in a two-stage approach for computing dominant Floquet multipliers, where global approximation is combined with local corrections. Correspondingly, we also propose two novel characterizations of left eigenvectors. Finally, from the nonlinear eigenvalue problem formulation, we derive computationally tractable expressions for derivatives of Floquet multipliers with respect to parameters, which are beneficial in the context of stability optimization. Several numerical examples show the efficacy and applicability of the presented results.

中文翻译：

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基于频谱的时滞系统稳定性分析和稳定性

我们开发了一种基于特征值的方法，用于具有多个延迟和周期系数矩阵的线性系统的稳定性评估和稳定性。延迟和周期被假定为相称的数，这样 Floquet 乘数可以被表征为单一性算子的特征值和有限维非线性特征值问题的解，其中特征矩阵的评估涉及解决初始值问题. 我们证明了这种双重解释可以用于计算主要 Floquet 乘数的两阶段方法，其中全局近似与局部校正相结合。相应地，我们还提出了左特征向量的两个新特征。最后，从非线性特征值问题公式化，我们推导出 Floquet 乘数关于参数的导数的计算易处理的表达式，这在稳定性优化的背景下是有益的。几个数值例子显示了所呈现结果的有效性和适用性。