Weak and Semi-Contraction for Network Systems and Diffusively Coupled Oscillators
Weak and Semi-Contraction for Network Systems and Diffusively Coupled Oscillators
复制标题
网络系统和扩散耦合振荡器的弱收缩和半收缩
DOI：
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发表时间：
2020
影响因子：
6.8
通讯作者：
F. Bullo
中科院分区：
文献类型：
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作者：
Saber Jafarpour;Pedro Cisneros;F. Bullo
我们发展了收缩理论的两种推广，即半收缩理论和弱收缩理论。首先，利用半范数的概念，我们提出了半收缩理论的几何框架。我们引入矩阵半测度并表征它们的性质。我们证明矩阵的谱横坐标是加权半测度的下确界。对于动力系统，我们使用雅可比行列式的半测度来表征其轨迹的收缩特性。其次，对于弱收缩系统，我们证明了其轨迹渐进行为的二分法和收敛到均衡的新颖充分条件。第三，我们证明了双重契约系统（即弱契约和半契约系统）的每一条轨迹都会收敛到一个平衡点。最后，我们将我们的结果应用于各种重要的网络系统，包括仿射平均和仿射流系统、连续时间分布式原对偶算法以及扩散耦合动力系统的网络。对于扩散耦合系统，半收缩理论提供了同步的充分条件，通常比以前已知的测试更清晰。
We develop two generalizations of contraction theory, namely, semi-contraction and weak-contraction theory. First, using the notion of seminorm, we propose a geometric framework for semi-contraction theory. We introduce matrix semimeasures and characterize their properties. We show that the spectral abscissa of a matrix is the infimum over weighted semimeasures. For dynamical systems, we use the semimeasure of their Jacobian to characterize the contractivity properties of their trajectories. Second, for weakly contracting systems, we prove a dichotomy for the asymptotic behavior of their trajectories and novel sufficient conditions for convergence to an equilibrium. Third, we show that every trajectory of a doubly contracting system, i.e., a system that is both weakly and semi-contracting, converges to an equilibrium point. Finally, we apply our results to various important network systems, including affine averaging and affine flow systems, continuous-time distributed primal-dual algorithms, and networks of diffusively coupled dynamical systems. For diffusively coupled systems, the semi-contraction theory leads to a sufficient condition for synchronization that is sharper, in general, than previously known tests.