Set Invariance and Optimal Set Stabilization of Boolean Control Networks: A Graphical Approach

Abstract

This article investigates set invariance and optimal set stabilization of Boolean control networks (BCNs) from a novel graph-theoretical perspective. First, we establish the relationship between the control invariant sets and the strongly connected components of the state transition graph (STG), and propose a new algorithm to compute the largest control invariant subset. Second, both set stabilizability and time-optimal set stabilization via state feedback are characterized by a breadth-first tree of the STG. Though the prior two problems have been attempted by existing algebraic methods, the primary advantage of our alternative graphical approach is the enhanced computational efficiency. Third, we consider for the first time, the optimal set stabilization problem with general cost criteria, not limited to time optimality. A state feedback controller is constructed for general optimal set stabilization using a shortest-path tree of the STG. Finally, it is straightforward to extend the proposed approach to handle various constraints as well as other types of logical networks by slightly modifying the STG. Two biological networks are used to test the performance of our approach. Results show that in accordance with time complexity analysis, our algorithms can dramatically reduce the running time for the first two problems compared with existing work, and effectively solve the general optimal set stabilization problem.