A Control Theory for Boolean Monomial Dynamical Systems

Recently criteria for determining when a certain type of nonlinear discrete dynamical system is a fixed point system have been developed. This theory can be used to determine if certain events modeled by those systems reach a steady state. In this work we formalize the idea of a “stabilizable” discrete dynamical system. We present necessary and sufficient conditions for a Boolean monomial dynamical control system to be stabilizable in terms of properties of the dependency graph associated with the system. We use the equivalence of periodicity of the dependency graph and loop numbers to develop a new O(n ²logn) algorithm for determining the loop numbers of the strongly connected components of the dependency graph, and hence a new O(n ²logn) algorithm for determining when a Boolean monomial dynamical system is a fixed point system. Finally, we show how this result can be used to determine if a Boolean monomial dynamical control system is stabilizable in time O(n ²logn).