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.

中文翻译：

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网络系统和扩散耦合振荡器的弱和半收缩

我们发展了收缩理论的两种概括，即半收缩理论和弱收缩理论。首先，使用半范数的概念，我们提出了半收缩理论的几何框架。我们介绍了矩阵半度量并描述了它们的特性。我们证明矩阵的谱横坐标是加权半度量的下界。对于动力系统，我们使用其雅可比的半测度来表征其轨迹的收缩特性。其次，对于弱收缩系统，我们证明了其轨迹的渐近行为和收敛到平衡的新充分条件的二分法。第三，我们证明了双收缩系统的每一条轨迹，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、连续时间分布式原始对偶算法以及扩散耦合动力系统网络。对于扩散耦合系统，半收缩理论导致同步的充分条件，通常比以前已知的测试更清晰。