  This paper investigates the finite-horizon optimal control (FHOC) problem of
Boolean control networks (BCNs) from a graph theory perspective. We first
formulate two general problems to unify various special cases studied in the
literature: (i) the horizon length is $\textit{a priori}$ fixed; (ii) the
horizon length is unspecified but finite for given destination states. Notably,
both problems can incorporate time-variant costs, which are rarely considered
in existing work, and a variety of constraints. The existence of an optimal
control sequence is analyzed under mild assumptions. Motivated by BCNs' finite
state space and control space, we approach the two general problems in an
intuitive and efficient way under a graph-theoretical framework. A weighted
state transition graph and its time-expanded variants are developed, and the
equivalence between the FHOC problem and the shortest path problem in specific
graphs is established rigorously. Two custom algorithms are developed to find
the shortest path and construct the optimal control sequence with lower time
complexity, though technically a classical shortest-path algorithm in graph
theory is sufficient for all problems. Compared with existing algebraic
methods, our graph-theoretical approach can achieve state-of-the-art time
efficiency while targeting the most general problems. Furthermore, our approach
is 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 a
benchmark example to validate the effectiveness of our approach, and the
results of two tasks show that our approach can dramatically reduce the running
time.
