Koopman Theory for Boolean Networks

Networks of agents with logical states, namely Boolean networks, arise in various application domains including biology, computer networks, and social networks. The representation and control of Boolean networks have attracted much attention in recent years. In a parallel line of research, Koopman developed an operator view of nonlinear dynamical systems, which shows that, by making use of observable functions, every nonlinear dynamics can be represented as a (possibly infinite dimensional) linear system. In this paper, we present a Koopman theory for Boolean networks. We introduce algebraic operations for logical functions over semirings of binary numbers, defining a semimodule of Boolean functions. The classical Koopman operator and Koopman theory are then established for Boolean networks, where the key ingredient to a linear representation is shown to be Koopman invariance. Next, we establish a necessary and sufficient condition for shaping the closed-loop dynamics via feedback into any desired form for Boolean control networks under the Koopman representation, and we develop a feedback control synthesis algorithm to solve this feedback shaping problem. Finally, feedback shaping is applied to a real-world gene regulatory network, and it is demonstrated that the dynamics of such a network may be arbitrarily manipulated by means of feedback control.