论文标题
时间尺度动态系统的矩阵测量，稳定性和收缩理论
Matrix measures, stability and contraction theory for dynamical systems on time scales
论文作者
论文摘要
本文涉及研究动态系统在时间尺度上发展的稳定性。我们首先{正式将矩阵测量的概念按时间尺度进行，证明了它们的某些关键属性，并利用此概念在时间尺度上同时研究线性和非线性动力学系统。}具体而言，我们首先要考虑线性时间变化的系统，对于这些系统，我们证明了由Coppel造成的上限的时间尺度。我们利用这种上限为线性时变系统提供稳定性和输入对状态的稳定条件。 {然后，我们考虑在时间尺度上考虑非线性时变动力系统，}为解决方案的收敛建立了足够的条件。最后，将我们的结果与Lyapunov函数的存在联系起来之后，我们利用方法来研究某些流行病动力学和复杂的网络。对于前者，我们在时间尺度上为SIQR模型的参数提供了足够的条件，以确保其溶液会融合到无疾病的溶液中。对于后者，我们首先提供了足够的条件来固定复杂的时间尺度网络的可控性，然后使用此条件来研究某些集体意见动态。理论结果与模拟相辅相成。
This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first {formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study both linear and nonlinear dynamical systems on time scales.} Specifically, we start with considering linear time-varying systems and, for these, we prove a time scale analogous of an upper bound due to Coppel. We make use of this upper bound to give stability and input-to-state stability conditions for linear time-varying systems. {Then, we consider nonlinear time-varying dynamical systems on time scales and} establish a sufficient condition for the convergence of the solutions. Finally, after linking our results to the existence of a Lyapunov function, we make use of our approach to study certain epidemic dynamics and complex networks. For the former, we give a sufficient condition on the parameters of a SIQR model on time scales ensuring that its solutions converge to the disease-free solution. For the latter, we first give a sufficient condition for pinning controllability of complex time scale networks and then use this condition to study certain collective opinion dynamics. The theoretical results are complemented with simulations.
