Pinning Synchronization of Large-Scale Boolean Networks

Synchronization of large-scale Boolean networks (BNs) is studied in this article. Using semi-tensor product of matrices, the algebraic structure of all subnetworks' relationships is constructed. A necessary and sufficient condition is obtained for synchronization of large-scale BNs. For those asynchronous large-scale BNs, pinning control is first applied to achieve the control objective, including choosing the subnetworks to be controlled, by designing the corresponding feedback control according to the subnetworks' states independently. In this article, for each pinned subnetwork, the related adding approach of feedback control is fixed to be Boolean addition, which is similar to the system addition. An algorithm is presented for getting a desired structure matrix, under which, the solvability of pinning controls including the exact pinning subnetworks and the corresponding feedback control design is addressed to guarantee the synchronization. An example modeled by biochemical oscillator is given to show how to use the results obtained in this article and also show the efficiency of pinning control.