Stabilizing Boolean networks by optimal event-triggered feedback control

Abstract

In this paper, we focus on the topic of stabilizing the Boolean control network (BCN) by an optimal event-triggered feedback control. By routinely transforming the BCN into its algebraic form, constructing the (reverse) weighted digraph and the hypergraph for the BCN, applying the shortest path algorithm to the hypergraph, we obtain an optimal event-triggered control strategy that can stabilize the BCN starting from any given initial state to a priorly-specified approachable equilibrium and simultaneously can minimize a quality index combining control time with control variability. To avoid the case where the obtained optimal strategy is dependent on the initial state, we establish necessary and sufficient conditions for finding a unified control law which can steer the BCN starting from all initial states to the given equilibrium with a minimal quality index. Furthermore, all the existence results are extended further to the BCNs under the control laws of event-triggered delayed state feedback, event-triggered output feedback, and the event-triggered delayed output feedback. Particularly, the control law of event-triggered delayed feedback can cope with the optimization problem that the unified control law without delay cannot address. Additionally, we derive the necessary and sufficient condition under which the event-triggered state (resp., output) feedback control law degenerates into the conventional state (resp., output) feedback control law. Finally, we provide an illustrative example of biological significance to demonstrate the potential usefulness of the obtained analytical results.