A General Control Framework for Boolean Networks

This paper focuses on proposing a general control framework for large-scale Boolean networks (\texttt{BNs}). Only by the network structure, the concept of structural controllability for \texttt{BNs} is formalized. A necessary and sufficient criterion is derived for the structural controllability of \texttt{BNs}; it can be verified with $\Theta(n^2)$ time, where $n$ is the number of network nodes. An interesting conclusion is shown as that a \texttt{BN} is structurally controllable if and only if it is structurally fixed-time controllable. Afterwards, the minimum node control problem with respect to structural controllability is proved to be NP-hard for structural \texttt{BNs}. In virtue of the structurally controllable criterion, three difficult control issues can be efficiently addressed and accompanied with some advantages. In terms of the design of pinning controllers to generate a controllable \texttt{BN}, by utilizing the structurally controllable criterion, the selection procedure for the pinning node set is developed for the first time instead of just checking the controllability under the given pinning control form; the pinning controller is of distributed form, and the time complexity is $\Theta(n2^{{3d{\ast}}+2(n+m)2)$,} where $m$ and $d^\ast$ are respectively the number of generators and the maximum vertex in-degree. With regard to the control design for stabilization in probability of probabilistic \texttt{BNs} (\texttt{PBNs}), an important theorem is proved to reveal the equivalence between several types of stability. The existing difficulties on the stabilization in probability are then solved to some extent via the structurally controllable criterion.