Finite-time optimal pinning control and synchronization for partially interdependent networks

For partially interdependent networks composed of two subnetworks, the finite-time optimal pinning control problem is investigated. Among them, only a part of the nodes between the two subnetworks are interdependent on each other. In the network, the coupling relationship between any two nodes of the network is a continuous nonlinear function. Based on the pinning control, the optimal control theory, Kalman's controllability rank conditions, and introducing the Lagrange function and applying controllers to the partial nodes of the network, we propose some characterization indicators which ensure that the two subnetworks of partially interdependent networks can synchronize to the equilibrium points of their isolate systems, separately. Finally, some numerical simulations are performed on the partially interdependent networks including two small world subnetworks. The results show that the optimal pinning control method proposed in this article can greatly reduce the control costs and achieve the ideal synchronous status quickly.