Feedback Shaping for Boolean Networks

Boolean networks, as logical dynamical systems where the system states are Boolean variables, arise from applications in biology, computer networks, and social networks etc. In this paper, we present a framework to evaluate whether and how the closed-loop dynamics of a controlled Boolean network can be shaped into any prescribed form by state-feedback control. We refer to this problem as to the feedback shaping for Boolean networks. First of all, based on the linear representation of Boolean networks, we establish a necessary and sufficient rank condition for a controlled Boolean network to be feedback shapable or not. Next, we design an algorithm for the synthesis of closed-loop dynamics for a feedback shapable Boolean network, such that for any given controlled Boolean network and a desired closed-loop dynamics, one can always find a feedback control law so that the closed-loop dynamics is precisely realized.