Continuous-time distributed convex optimization on time-varying directed networks

We introduce a general class of continuous-time distributed control systems, where the control input to the dynamics of each agent relies on an observer that estimates the average state. The dynamics of these observers are nonlinear, but the agents only need to have access to local information to implement them. We show that under a general condition on the structure of the underlying time-varying directed graphs, the difference of the agents´ estimates and the true average is upper bounded. Using this result, we show that when we have a class P* weakly exponentially ergodic flow and the agent´s objective functions are differentiable with bounded gradients, any trajectory of the proposed continuous-time dynamics is globally asymptotically convergent to a minimizer. Finally, we demonstrate that the class P* weakly exponentially ergodic flow property can be achieved by assuming that the sequence of Laplacians are measurable, cut-balanced, and has a minimum instantaneous flow. As a by-product, we show that the proposed continuous-time dynamics for distributed convex optimization is convergent on any sequence of time-varying strongly connected directed graph.