Abstract: Dynamical systems on large-scale networks appear in numerous real-world applications, including traffic and power infrastructures, social interactions, chemical and biological systems, and neural networks. In most of these applications, dynamic networks feature large numbers of components and complicated patterns of interaction. Contraction theory is a scalable stability framework that defines stability incrementally between two arbitrary trajectories. However, the applicability of classical contraction theory to real-world networks is limited as these networks are often not strictly contracting due to conservation laws or symmetries.

In this talk, we develop two extensions of contraction theory, namely weak contraction and semi-contraction. For weak-contracting systems, we provide a dichotomy for the asymptotic behavior of their trajectories and propose sufficient conditions for their convergence to equilibria. For semi-contracting systems, we study the convergence of their trajectories to affine invariant subspaces. Finally, we apply our results to two essential classes of network systems: i) continuous-time distributed primal-dual algorithms over networks and ii) diffusively coupled dynamical systems.

Bio: Saber Jafarpour is a postdoctoral research fellow in the Department of Electrical and Computer Engineering at the Georgia Institute of Technology. Before that, he was a postdoctoral researcher in the Center for Control, Dynamical Systems, and Computation at UC Santa Barbara. He received his Ph.D. in Mathematics and Statistics from Queen’s University.