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

## TL;DR

This paper presents a graph-theoretical framework for finite-horizon optimal control of Boolean control networks, enabling efficient solutions for complex problems including time-variant costs and various constraints, validated on a genetic network example.

## Contribution

It introduces a unified graph-based approach for FHOC of BCNs, capable of handling general problems with time-variant costs and improving computational efficiency over existing methods.

## Key findings

- Developed weighted state transition graphs and algorithms for shortest path problems.
- Proved equivalence between FHOC and shortest path problems in specific graphs.
- Validated approach on E. coli genetic network, showing significant time reduction.

## Abstract

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.