Output Regulation of Boolean Control Networks Under Probabilistic Outputs

In this article, the output regulation problem (ORP) of Boolean control networks (BCNs) is studied via the semitensor product of matrices, with consideration of probabilistic outputs from the reference Boolean networks (BNs). We first define the concept of mean square error (MSE) between outputs of two BNs, establishing several theoretical results on the solvability of the ORP. Subsequently, state feedback controllers are designed to address the ORP of BCNs under different scenarios. In the first scenario, ORP is addressed by developing control strategies that ensure the system's output remains within a predefined MSE threshold. In the second scenario, we focus on achieving the minimum MSE through the optimization of control strategies. The final scenario extends the second by conducting a detailed analysis to maintain this minimum MSE uniformly across all state pairs. To conclude, several numerical examples are provided to verify the effectiveness of the proposed theoretical results.