This article explores the influence of stochastic function perturbations (SFPs) on the output tracking of Boolean control networks (BCNs). First, by establishing the transition probability matrix of BCNs subject to SFPs, a criterion is presented to verify whether or not the state transition of BCNs is affected by SFPs. Second, when the state transition of BCNs is influenced by SFPs, a perturbed parameterized set is constructed via the information of original BCNs and SFPs. Based on the perturbed parameterized set, several necessary and sufficient conditions are derived for the robust output tracking of BCNs subject to SFPs, including keeping finite-time output tracking and reducing to asymptotic output tracking. Finally, the obtained theoretical results are applied to a Boolean model of lac operon in the Escherichia coli.

Edge perturbations (EPs) are often induced by disease-causing mutations, which include edge removal and edge sign switch. This article explores the robust stability of Boolean networks (BNs) with these two kinds of EPs. At first, both edge removal perturbation and edge sign switch perturbation are converted to multibit function perturbations (FPs). After that, the parameterized reachability matrix is constructed to verify the robust stability of BNs with multibit FP. As applications, several new criteria are established for the robust stability of BNs subject to edge removal perturbation and edge sign switch perturbation. Finally, the results are demonstrated by the Boolean models of p53 pathways network and Toll pathway of Drosophila signaling pathway network.

The law of conservation of mass, represented in Boolean networks (BNs) as total-activity conservation, is one of the typical properties of biological networks. This article analyzes the total-activity conservation of BNs based on the algebraic state-space representation (ASSR) approach. First, the total-activity-conservative matrix is defined and a matrix-based criterion is proposed to verify the total-activity conservation of BNs. Meanwhile, when function perturbation is considered, robust total-activity conservation is investigated. Second, by means of the pseudo-Boolean function generated by total-activity conservation, a constructive design procedure of the Boolean dynamics is given to achieve the total-activity conservation. Third, the total-activity conservation of switched Boolean networks (SBNs) under arbitrary switching signal is studied, which together with network aggregation achieves the total-activity conservation of large-scale BNs.

Robust controllability and stabilization are two fundamental issues in modern control theory. This article investigates the robust controllability and stabilization of Boolean control networks (BCNs) subject to stochastic function perturbations (SFPs). First, by defining the concept of the weighted path and constructing the perturbed position index matrix, the robustness analysis of reachability is divided into two cases. According to these two cases, several criteria are proposed for the robust reachability of BCNs with SFPs. Second, based on the robust reachability, the robust controllability and state-feedback stabilization of BCNs with SFPs are further studied. Finally, the validity of the obtained results is supported by examples.

This article analyzes the function perturbation impact on the finite-time stability and stabilization of the probabilistic Boolean networks (PBNs). First, the concept of stability in the distribution of PBNs is divided into two disjoint concepts, that is, finite-time stability with probability one (FTSPO) and asymptotical stability with probability one (ASPO), and a new criterion is proposed for the verification of ASPO. Second, by constructing a parameterized set, it is shown that PBNs subject to function perturbation keep FTSPO if and only if the perturbed point does not belong to the parameterized set, while PBNs become ASPO if and only if the perturbed point belongs to the parameterized set. Third, as an application of perturbed stability analysis, the robust state-feedback stabilization is discussed for probabilistic Boolean control networks (PBCNs) with function perturbation. Finally, the obtained results are applied to a WNT5A network and lac operon in the Escherichia coli, respectively.