Global robust stability and stabilization of Boolean network with disturbances

Based on semi-tensor product of matrices, global robust stability and stabilization of Boolean network (BN) with disturbances are investigated. Firstly, the definition of global robust stability for BN with disturbances is presented. By using the algebraic state space representation of disturbed BN, some necessary and sufficient criteria are obtained to ensure the global robust stability with respect to (w.r.t.) a fixed point or w.r.t. a limit cycle. In contrast, if a given disturbed BN is not globally robust stable w.r.t. a fixed point or w.r.t. a limit cycle, system can achieve stability by a matrix transformation technique. However, it is a difficult task to find such a suitable matrix transformation. In this paper, a pinning state feedback control design is proposed to find a suitable transformation. The obtained results are well illustrated by numerical example.