The analysis of local convergence of Boolean networks with state-based disturbances

This paper discusses the local convergence of Boolean networks (BNs) with state-based disturbances by pinning control. Under the framework of semi-tensor product (STP) of matrices, the algebraic expression of a Boolean network (BN) can be obtained. Then, the definition of local convergence is given. Note that compared with the stability, the time is limited to no more than n steps rather than 2n when considering the local convergence. By analyzing the algebraic expression, some necessary and sufficient conditions for local convergence were derived. Moreover, how to choose the pinning-nodes was analyzed, and pinning controllers are designed such that systems with state-based disturbances achieve locally convergence. Finally, the proposed method is well demonstrated by some examples.