Time-varying convex optimization for double-integrator dynamics over a directed network

In this paper, a time-varying distributed convex optimization problem is studied for a multi-agent system with double-integrator dynamics over a fixed directed network. The objective is to minimize a team cost function formed by the sum of local time-varying cost functions, each of which is known to only an individual agent, through local interaction. Here the optimal point is time-varying and creates an optimal trajectory. Sufficient conditions are derived to guarantee that all agents reach a consensus in a finite time while minimizing the team cost function.