This paper investigates the controllability and optimal control of higher-order incomplete Boolean control networks (BCNs) with impulsive effects by using the semi-tensor product of matrices. Firstly, the incomplete logical system is expressed as an algebraic form, based on which several necessary and sufficient conditions for the controllability are presented. Then, the Mayer-type optimal control issue is studied and the optimal control design algorithms are established. At last, the study of illustrative examples shows the effectiveness of the obtained new results.

This paper investigates robust set stability for the switched Boolean networks (SBNs) with arbitrary switching signal affected by one-bit function perturbation (OBFP). Firstly, the dynamics of these networks are converted into algebraic forms utilizing semi-tensor product (STP) method. Secondly, OBFP impact on the set stability of SBNs is divided into four cases. Then, by constructing a state set and defining an index vector, several necessary and sufficient conditions are provided to detect whether an SBN with arbitrary switching signal is still stable to the given set after OBFP. Finally, a biological example is proposed to demonstrate the effectiveness of the obtained theoretical results.

This article concentrates on the function perturbation impact on robust stability and robust stabilization of Boolean networks with disturbance inputs (DBNs). First, using the semi-tensor product (STP) of matrices, the algebraic representation of DBNs with function perturbation is given. Second, a state set is determined to detect the robust stability of DBNs subject to function perturbation. The result shows that the robust stability of DBNs remains unchanged if and only if the perturbed point is not in the constructed state set. Third, DBNs with control inputs (DBCNs) are considered, and several criteria to verify whether DBCNs with function perturbation can still maintain robust stabilization under a given state feedback stabilizer are presented. Finally, two examples are provided to illustrate the validity of the theoretical results.

This paper studies the optimal control problem for a class of networked evolutionary games (NEGs) with bankruptcy risk. Firstly, an algebraic expression is formulated for the given NEGs via the semi-tensor product. Then, the optimal control problem is discussed, and a state feedback controller is designed to maximize the long-run average payoff of the game. Finally, an illustrative example is presented to show the effectiveness of the obtained new results.