In genetic regulatory networks (GRNs), gene mutations often occur in a stochastic manner. As an important model of GRNs, gene mutations of Boolean networks are always described as function perturbations. This article studies the robust stability and stabilization of Boolean networks with stochastic function perturbations. A kind of parameterized set is constructed, and it is revealed that under the stochastic function perturbations, the property of finite-time stability remains unchanged when the perturbed set and the parameterized set are disjoint. In addition, it is proved that the finite-time stability is reduced to stability in distribution when the intersection of perturbed set and complement set of parameterized set is nonempty. As an application, the robust stabilization problem of Boolean control networks with stochastic function perturbations is discussed, and several necessary and sufficient conditions are presented for the robustness of feedback stabilizers. Finally, the obtained results are used to study the Drosophila melanogaster segmentation polarity gene network and the lac operon in the bacterium Escherichia coil.

中文翻译：

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具有随机函数摄动的布尔网络的稳定性和稳定性的鲁棒性

在遗传调控网络（GRN）中，基因突变通常以随机方式发生。作为GRN的重要模型，布尔网络的基因突变总是被描述为功能扰动。本文研究具有随机函数摄动的布尔网络的鲁棒稳定性和稳定性。构造了一种参数化集，揭示了在随机函数扰动下，当扰动集与参数化集不相交时，有限时间稳定性的性质保持不变。此外，证明了当扰动集与参数集的补集的交集为非空时，有限时间稳定性降低为分布的稳定性。作为应用，讨论了具有随机函数扰动的布尔控制网络的鲁棒镇定问题，并给出了反馈稳定器鲁棒性的若干必要条件和充分条件。最后，将获得的结果用于研究果蝇果蝇分割极性基因网络和大肠埃希氏菌属细菌中的lac操纵子。