# Optimization-Based Approaches to Control of Probabilistic Boolean Networks
## 6 Open Problems in Control Theory of Probabilistic Boolean Networks  
We have explained our previously proposed methods, and we have compared these methods with other existing methods. In this section, we suggest open problems in control theory of PBNs. Here, we consider not only optimal control methods but also other control methods.  
Although our previously proposed methods can handle large-scale PBNs, these methods are not scalable. Developing scalable control methods is important. Then, decomposition of networks is important (see, e.g., [50; 51]). For BNs, we consider only one network structure (directed graph) such as Figure 1. On the other hand, for PBNs, we must consider a set of directed graphs. Hence, the problem of decomposing given networks in a PBN is more difficult than that in a BN. This is one of the challenging problem in control of PBNs. Moreover, the optimal control problems for PBNs obtained from decomposed graphs can be solved fast, but their solutions may not be an optimal solution for the optimal control problem of a given PBN. As a result, the optimality of the cost function is degraded. Now, some analysis of the approximation accuracy is required.  
It is also important to consider what can be obtained from graph structure alone. In [22], controllability of BNs is characterized using a time sequence of graph structure. In [16], singleton attractors (fixed points) of BNs are characterized from graph structure only. In [52; 53], a feedback vertex set of a graph is focused. A feedback vertex set of a graph is a set of vertices whose removal results an acyclic graph. For a given graph, several feedback vertex sets can be obtained, but in particular, a minimum feedback vertex set in which the number of vertices is minimum is important. A minimum feedback vertex set is also not given uniquely. It was shown in [52; 53] that in the system given by a differential equation, a certain property of all states can be characterized by analyzing the states relating to a certain feedback vertex set. However, the related results for BNs and PBNs have not been obtained so far.  
In our previously proposed methods explained in this paper, a controller must be implemented by the personal computer that can solve a polynomial optimization problem and an integer programming problem (of course, actuators for interacting genes are necessary, but this topic is out of scope of this review paper). We may consider the case where a controller must be described by a certain Boolean function. In such a case, the algebraic representation using the STP and the matrix-based representation [54; 55] are useful. Details are one of the open problems.  
Since BNs and PBNs are synchronous models, an extension to asynchronous models is important. Then, it is appropriate to utilize a Petri net representation. A Petri net is well known as a model of asynchronous behavior (see, e.g., [56] for further details), and is also used as a model of biological networks (see, e.g., [9; 57]). In [58], the authors have proposed a Petri net representation for modeling asynchronous BN. In the case of PBNs, a stochastic Petri net may be used as a model of asynchronous PBNs, but further discussion is required.  
Finally, the state and the control input in BNs and PBNs are binary. There is a possibility that biological behavior is not precisely modeled (see, e.g., [11]). Then, it may be appropriate that some of states/inputs are given by continuous-valued variables. A dynamical system with binary variables and continuous variables is called a hybrid dynamical system (see, e.g., [59]). Further research on modeling and control methods is needed.  
Thus, in control theory of PBNs, there are many remaining open problems. Further development of many control methods is required.