Optimal Pinning Control for Synchronization over Temporal Networks

In this paper, we address the finite time synchronization of a network of dynamical systems with time-varying interactions modeled using temporal networks. We synchronize a few nodes initially using external control inputs. These nodes are termed as pinning nodes. The other nodes are synchronized by interacting with the pinning nodes and with each other. We first provide sufficient conditions for the network to be synchronized. Then we formulate an optimization problem to minimize the number of pinning nodes for synchronizing the entire network. Finally, we address the problem of maximizing the number of synchronized nodes when there are constraints on the number of nodes that could be pinned. We show that this problem belongs to the class of NP-hard problems and propose a greedy heuristic. We illustrate the results using numerical simulations.