Toward Stabilizable Large-Scale Boolean Networks by Controlling the Minimal Set of Nodes

Abstract

This paper studies <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">structural</i> controllers and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributed</i> pinning controllers for the global stabilization of Boolean networks (BNs) by integrating the information on their network structures with nodal dynamics. The main contribution is that several computationally efficient procedures are presented to reduce the number of controlled nodes and to determine a minimal set of controlled nodes without using the brute-force searches. The primary objective is to identify a minimal set of nodes that need to be controlled in the structural controllers for the strong structural stabilization of BNs when network structures is available yet nodal dynamics are unknown. To this end, a theorem shows this minimum controlled node problem can be addressed by seeking the minimum feedback vertex set of network structure. The subsequent part concentrates on designing distributed pinning controllers that merely rely on the node-to-node information exchanges for the global stabilization of BNs with the full knowledge of the nodal dynamics. Several sufficient conditions are provided by utilizing the irreducibility and activation-inhibition network structures to reduce the conservatism. Notably, we claim that, for regulatory BNs without positive cycles, the minimal set of pinned nodes can be determined with a linear amount of time subject to the total number of logical operators in the nodal dynamics. Finally, the effectiveness of these results are validated by three case studies.