Infinite-Horizon Optimal Control of Switched Boolean Control Networks with Average Cost: An Efficient Graph-Theoretical Approach

TL;DR

This paper introduces a graph-theoretical method for efficiently solving the infinite-horizon average-cost optimal control problem in large-scale switched Boolean control networks, significantly reducing computational complexity.

Contribution

It develops a novel graph-based framework and algorithm that outperforms existing methods in speed for large networks, using the optimal state transition graph and minimum mean cycle problem.

Findings

The proposed method is hundreds to thousands of times faster than existing approaches.

It successfully applies to a 16-node leukemia signaling network as a benchmark.

The approach effectively reduces the control problem to a minimum mean cycle problem in a specialized graph.

Abstract

This study investigates the infinite-horizon optimal control problem for switched Boolean control networks with an average-cost criterion. A primary challenge of this problem is the prohibitively high computational cost when dealing with large-scale networks. We attempt to develop a more efficient and scalable approach from a graph-theoretical perspective. First, a weighted directed graph structure called the $\textit{optimal state transition graph}$ (OSTG) is established, whose edges encode the optimal action for each one-step transition between states reachable from a given initial state subject to various constraints. Then, we reduce the infinite-horizon optimal control problem into a minimum mean cycle (MMC) problem in the OSTG. Finally, we develop a novel algorithm that can quickly find a particular MMC by resorting to Karp's algorithm in graph theory and construct afterward an optimal switching and control law based on state feedback. Time complexity analysis shows that our algorithm can outperform all existing methods in terms of time efficiency. A 16-node signaling network in leukemia is used as a benchmark to test its effectiveness. Results show that the proposed graph-theoretical approach is much more computationally efficient: it runs hundreds or even thousands of times faster than existing methods.