Specific target pinning control of complex dynamical networks based on semidefinite programming strategy

Pinning control of complex networks aims to force the states of all nodes to reach the desired goal by controlling a few nodes in the network. The easiness of a pinning strategy can be measured by the minimum eigenvalue λ1 of the specific matrix L+Γ, where L is the Laplacian matrix of the underlying network and Γ is a diagonal matrix with Γi,i=1 if node i is pinned. A larger value λ1 indicates a smaller coupling strength. We here are interested in two key questions on this issue. The first one is, what is the minimum number of pinned nodes to achieve system synchronization when individual node dynamics and the coupling strength of the network are given? We show an upper bound of the minimum eigenvalue λ1, by which a lower bound of the number of pinned nodes is presented. The second one is, which nodes should be pinned such that the system achieves synchronization much easier if a potential pinning target node set S0 (instead of the whole node set) and the number of pinned nodes l (l