Fast-Time Stability of Temporal Boolean Networks

In real systems, most of the biological functionalities come from the fact that the connections are not active all the time. Based on the fact, temporal Boolean networks (TBNs) are proposed in this paper, and the fast-time stability is analyzed via semi-tensor product (STP) of matrices and incidence matrices. First, the algebraic form of a TBN is obtained based on the STP method, and one necessary and sufficient condition for global fast-time stability is presented. Moreover, incidence matrices are used to obtain several sufficient conditions, which reduce the computational complexity from <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(n2^{n})$ </tex-math></inline-formula> (exponential type) to <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(n^{4})$ </tex-math></inline-formula> (polynomial type) compared with the STP method. In addition, the global fast-time stabilization of TBNs is considered, and pinning controllers are designed based on the neighbors of controlled nodes rather than all the nodes. Finally, the local fast-time stability of TBNs is considered based on the incidence matrices as well. Several examples are provided to illustrate the effectiveness of the obtained results.