Stability Analysis and Control of Probabilistic Logical Dynamical Systems: Theory and Applications

With the increasing complexity and uncertainty in networked systems, probabilistic logical networks have become important mathematical models for describing systems with logical interactions and stochastic switching mechanisms, such as genetic regulatory networks and statebased evolutionary games. By means of the semi-tensor product (STP) of matrices, logical dynamic systems admit equivalent algebraic representations in finite-dimensional linear spaces, which provide a unified framework for rigorous analysis. However, systematic algebraic characterizations for set stability, stabilization control, structural decomposition of large-scale systems, and synchronization in distribution remain insufficiently developed for probabilistic logical networks. This thesis establishes an algebraic and iterative framework to address stability and control problems for several classes of probabilistic logical networks. First, we investigate the set stability problem of probabilistic mix-valued logical networks. An invariant subset theory under the STP representation is developed, and a matrix-based iterative procedure is constructed to compute the largest invariant subset contained in a prescribed set. Based on this construction, a necessary and sufficient algebraic criterion is derived for set stability from a given initial set to a target set with probability one. The obtained results provide a unified state-space characterization of stability for stochastic logical systems and are further applied to analyze the stability of recurrent state equilibria in state-based games, offering a novel state-space-based analytical method for evolutionary game dynamics. Second, we study the set stabilization problem of probabilistic Boolean control networks. A novel iterative framework is developed to compute the largest control invariant subset contained in a target set. On this basis, an algebraic construction of state feedback controllers is established to achieve global set stabilization with probability one. The proposed design avoids explicit rule enumeration and transition-table manipulations, yielding a structurally transparent and mathematically tractable stabilization framework. The developed theory is further applied to statebased games with control, where stabilization of recurrent state equilibria is characterized within the same algebraic setting. Third, to cope with the complexity of large-scale probabilistic Boolean networks, a network aggregation approach under general aggregation structures is developed. By decomposing the original network into interconnected subnetworks and deriving algebraic expressions for aggregated dynamics, tractable stability conditions are obtained for large-scale systems. This structural decomposition significantly reduces computational complexity while preserving essential dynamical properties. Finally, we investigate the problem of approximate synchronization in distribution for coupled probabilistic Boolean networks. A distribution-based notion of approximate synchronization is introduced, and the approximate behaviors of the associated probability transition matrices are analyzed. A verifiable algebraic criterion for distributional approximate synchronization is established, providing a theoretical foundation for studying collective behaviors in stochastic logical networks. Overall, this thesis establishes a unified algebraic and computational framework for stability analysis, stabilization control, large-scale decomposition, and synchronization of probabilistic logical networks, and demonstrates its applicability to game theory and other complex systems.