Output Tracking of Periodically Time-Varying Boolean Networks: State-Flipped Control and Q-Learning Approaches

This article investigates the output tracking problem for periodically time-varying Boolean networks (PTVBNs), motivated by rhythmic gene regulation and cyclic operating regimes in discrete systems. In such networks, the update rules change periodically, which makes tracking analysis and controller synthesis challenging, especially when only a subset of state components can be manipulated. A state-flipped control strategy is employed to address this challenge, enabling the modification of multiple nodes' states between binary values. Matrix-based representations are introduced to formalize both the output tracking problem and the state-flipped control mechanism. The article first develops an algebraic approach for analyzing output tracking in PTVBNs, establishing a comprehensive criterion for trackability. For synthesis with reduced model dependence, a model-free reinforcement learning formulation is further introduced. A two-level $Q$ -learning scheme is employed to identify the minimal state-flipping set required to successfully achieve output tracking. The effectiveness of the theoretical results is validated through extensive simulations on two biological systems: a repressilator model and a ten-node cell cycle network.