Necessary and Sufficient Conditions on Pinning Stabilization for Stochastic Boolean Networks

In this paper, the stabilization problem of the Boolean network (BN) with stochastic disturbances via pinning control has been investigated. The necessary and sufficient conditions are given for robust stabilization of a BN with stochastic disturbances. Then, pinning control is considered to stabilize a BN with stochastic disturbances. An algorithm is given to obtain a new stable system and the pinning control, including the pinned nodes, control design, and control adding, is also solved. Finding the minimal number of pinned nodes is further analyzed. The necessary and sufficient conditions are obtained for the solvability of pinning control, based on which some matrices sets are constructed which leads to the necessary and sufficient conditions of pinning $t$ nodes. Furthermore, an algorithm is introduced to search the minimal number of pinned nodes and what exactly they are, which will reduce the computational burden. Examples are given to illustrate the efficiency of the obtained results.