Identification and Control of Two Classes of Boolean Networks and the Applications in Game Theory

Abstract Boolean networks (BNs), also referred to as two-valued logical dynamical systems, provide a powerful discrete dynamical framework for modeling and analyzing complex systems in biology, engineering, and computer science. They also serve as an effective representation for bi-strategy evolutionary dynamics in finite-player games. Evolutionary game theory on complex networks has emerged as an important framework for studying collective behavior in systems with strategic interactions. In particular, the introduction of higher-order interactions via heterogeneous hypergraphs significantly enhances the modeling capability of evolutionary games. Evolutionary games on heterogeneous hypergraphs can be formulated as multi-valued logical dynamical systems. By resorting to the semi-tensor product (STP) of matrices, logical dynamical systems can be equivalently transformed into discrete-time linear algebraic systems. This algebraic representation also provides a natural bridge between BNs and evolutionary game dynamics. Based on that, this thesis develops an algebraic framework for the identification and control of BNs and switched Boolean networks (SBNs), and further extends it to heterogeneous hyper-networked evolutionary games (HNEGs). The main contributions are summarized as follows: • The identification problem for SBNs is investigated under general switching structures, where switching signals affect both the state evolution and the corresponding output behavior. The reachability and observability properties of SBNs are first analyzed, and corresponding conditions are established. Based on these properties, criteria for SBN identification under both singlesample and multi-sample scenarios are derived. By incorporating temporal information, the exact correspondence between system states and outputs is established, and efficient algorithms are developed for the identification procedure. Interestingly, an SBN may remain identifiable even when some of its subnetworks are unidentifiable. • A novel control objective, termed recurrent output tracking (ROT), is introvi ABSTRACT vii duced for BNs and Boolean control networks (BCNs). This formulation relaxes conventional tracking requirements by allowing system outputs to intermittently and infinitely track a desired reference signal. Algebraic solvability conditions are derived, and constructive state feedback controllers are designed. Moreover, the periodicity properties of BNs and BCNs are analyzed, showing that the classical tracking problem is a special case of the recurrent output tracking problem (ROTP). • The ROTP is extended to switched Boolean control networks (SBCNs). By introducing the concepts of recurrent states and reachable sets, necessary and sufficient conditions are established for the solvability of the ROTP for SBCNs. An efficient algorithm is further developed to design state feedback controllers and switching signals such that the system can recurrently track a desired reference output. This highlights the fundamental role of switching in determining system reachability and ROT performance. • Algebraic expression of heterogeneous HNEGs is established via the STP method, which overcomes the dimensional inconsistency of payoff functions and captures genuine multi-player interactions induced by hyperedges. By adding control players to the game, a controlled heterogeneous HNEG is introduced, and the strategy optimization problem is reformulated as a minimum-time set controllability problem. Necessary and sufficient conditions are derived to guarantee that all players’ payoffs simultaneously reach and remain above the prescribed threshold. Furthermore, a state feedback controller is designed to achieve the objective in minimum time.