  In this paper, a deep reinforcement learning based method is proposed to
obtain optimal policies for optimal infinite-horizon control of probabilistic
Boolean control networks (PBCNs). Compared with the existing literatures, the
proposed method is model-free, namely, the system model and the initial states
needn't to be known. Meanwhile, it is suitable for large-scale PBCNs. First, we
establish the connection between deep reinforcement learning and optimal
infinite-horizon control, and structure the problem into the framework of the
Markov decision process. Then, PBCNs are defined as large-scale or small-scale,
depending on whether the memory of the action-values exceeds the RAM of the
computer. Based on the newly introduced definition, Q-learning (QL) and double
deep Q-network (DDQN) are applied to the optimal infinite-horizon control of
small-scale and large-scale PBCNs, respectively. Meanwhile, the optimal state
feedback controllers are designed. Finally, two examples are presented, which
are a small-scale PBCN with 3 nodes, and a large-scale one with 28 nodes. To
verify the convergence of QL and DDQN, the optimal control policy and the
optimal action-values, which are obtained from both the algorithms, are
compared with the ones based on a model-based method named policy iteration.
Meanwhile, the performance of QL is compared with DDQN in the small-scale PBCN.
