网络系统的弱压缩和半压缩与扩散耦合振子
Weak and Semi-Contraction for Network Systems and Diffusively-Coupled Oscillators
摘要：
我们发展了压缩理论的两个推广，即半压缩理论和弱压缩理论。首先，利用半范数的概念，我们提出了半压缩理论的几何框架。引入了矩阵半测度并刻画了它们的性质。我们证明了矩阵的谱横坐标是加权半测度上的下确界。对于动力系统，我们利用其雅可比矩阵的半测度来刻画其轨迹的收缩性。其次，对于弱收缩系统，我们证明了其轨迹的渐近行为的二分法和收敛到平衡点的新的充分条件。第三，我们证明了双收缩系统的每个轨迹，即，一个既弱又半收缩的系统，收敛到一个平衡点。最后，我们将我们的结果应用到各种重要的网络系统，包括仿射平均和仿射流系统，连续时间分布的原始-对偶算法，和扩散耦合动力系统的网络。对于扩散耦合系统，半收缩理论导致同步的充分条件，这是尖锐的，一般来说，比以前已知的测试。
英文摘要：
We develop two generalizations of contraction theory, namely, semi-contraction and weak-contraction theory. First, using the notion of semi-norm, we propose a geometric framework for semi-contraction theory. We introduce matrix semi-measures and characterize their properties. We show that the spectral abscissa of a matrix is the infimum over weighted semi-measures. For dynamical systems, we use the semi-measure 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.
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