A Pontryagin Maximum Principle for Multi–Input Boolean Control Networks⋆

A Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. Boolean networks have been studied extensively as models for simple artificial neural networks. Recently, Boolean networks gained considerable interest as models for biological systems composed of elements that can be in one of two possible states. Examples include genetic regulation networks, where the ON (OFF) state corresponds to the transcribed (quiescent) state of a gene, and cellular networks where the two possible logic states may represent the open/closed state of an ion channel, basal/high activity of an enzyme, two possible conformational states of a protein, etc. Daizhan Cheng developed an algebraic state-space representation for Boolean control networks using the semi–tensor product of matrices. This representation proved quite useful for studying Boolean control networks in a control-theoretic framework. Using this representation, we consider a Mayer-type optimal control problem for Boolean control networks. Our main result is a necessary condition for optimality. This provides a parallel of Pontryagin’s maximum principle to Boolean control networks.