Stabilization of Delayed Boolean Networks Using Constrained State Pinning Control

Pinning control is applied to ensure the stabilization of Boolean networks (BNs) with time delay parameter. The time delay parameter in this article follows an independent identical distribution. Using semi-tensor product of matrices, the considered BN with time delay is converted to a high dimensional switching BN and the switching signal is the time delay signal. Different from general switching BN, the structure matrices of all subsystems are independent with each other, structure matrices of all subsystems for the high dimensional switching BN depend on the original BN structure matrix. Pinning control is designed to guarantee the global stochastic stability of the considered system. Furthermore, the global stochastic stability of BN with time delay parameter is proved to be equivalent to the stability of BN without time delay parameter, which greatly reduces the computational complexity and simplifies the control design. For the considered BN with finite cycles, constrained pinning control is applied with limitation that only one state of each undesired cycle is under control. With this constraint, the minimal number of pinning control nodes is further investigated. Some algorithms are presented to obtain the new structure matrix, with which, pinning control can be solved by the obtained methods. Both numerical examples and biological example illustrate the effectiveness of the results.