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This paper investigates the finite-horizon optimal control (FHOC) problem ofBoolean control networks (BCNs) from a graph theory perspective. We firstformulate two general problems to unify various special cases studied in theliterature: (i) the horizon length is $\textit{a priori}$ fixed; (ii) thehorizon length is unspecified but finite for given destination states. Notably,both problems can incorporate time-variant costs, which are rarely consideredin existing work, and a variety of constraints. The existence of an optimalcontrol sequence is analyzed under mild assumptions. Motivated by BCNs' finitestate space and control space, we approach the two general problems in anintuitive and efficient way under a graph-theoretical framework. A weightedstate transition graph and its time-expanded variants are developed, and theequivalence between the FHOC problem and the shortest path problem in specificgraphs is established rigorously. Two custom algorithms are developed to findthe shortest path and construct the optimal control sequence with lower timecomplexity, though technically a classical shortest-path algorithm in graphtheory is sufficient for all problems. Compared with existing algebraicmethods, our graph-theoretical approach can achieve state-of-the-art timeefficiency while targeting the most general problems. Furthermore, our approachis the first one capable of solving Problem (ii) with time-variant costs.Finally, the Ara operon genetic network in $\textit{E. coli}$ is used as abenchmark example to validate the effectiveness of our approach, and theresults of two tasks show that our approach can dramatically reduce the runningtime.

Finite-Horizon Optimal Control of Boolean Control Networks: A Unified Graph-Theoretical Approach