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This paper is concerned with the complete parametric solutions to the generalized discrete Yakubovich-transpose matrix equation X − AXTB = CY. which is related with several types of matrix equations in control theory. One of the parametric solutions has a neat and elegant form in terms of the Krylov matrix, a block Hankel matrix and an observability matrix. In addition, the special case of the generalized discrete Yakubovich-transpose matrix equation, which is called the Karm- Yakubovich-transpose matrix equation, is considered. The explicit solutions to the Karm- Yakubovich-transpose matrix equation are also presented by the so-called generalized Leverrier algorithm. At the end of the paper, two examples are given to show the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Network synchronization, explicating typical collective behaviors of coupled systems, plays a crucial role in social production and life. This paper addresses the synchronization problem of drive–response singular Boolean networks (SBNs). The solvability of drive–response SBNs is investigated based on the matrix representation. In view of the existence and uniqueness of the solutions to drive–response SBNs, three types of concepts, synchronization, strong synchronization and weak synchronization, are put forward for the first time. By two new systems, a restricted BN and a switched restricted BN, which are constructed from the considered systems, several synchronization conditions are provided to deal with the circumstances of unique solutions and multiple solutions, respectively. Besides, the synchronous ratio is defined to characterize the synchronization capability of drive–response SBNs for the case of multiple solutions. Finally, several examples are given to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

SUMMARY The finite time horizon singular linear quadratic (LQ) optimal control problem is investigated for singular stochastic discrete-time systems. The problem is transformed into positive LQ one for standard stochastic systems via two equivalent transformations. It is proved that the singular LQ optimal control problem is solvable under two reasonable rank conditions. Via dynamic programming principle, the desired optimal controller is presented in terms of matrix iterative form. One simulation is provided to show the effectiveness of the proposed approaches. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper considers the problem of H∞ output feedback control with spectrum constraints for a class of uncertain stochastic systems. The purpose is the design of dynamic output feedback controllers such that the closed-loop system is asymptotically mean-square stable with a specified stability degree and an H∞-norm bound. First, the concept of spectrum for stochastic systems is redefined, and some characterizations about the stability degree are presented. Then, several equivalent conditions for the existence of H∞ dynamic output feedback controllers are obtained, and the explicit expression of a desired controller is provided. Furthermore, by using the cone complementarity linearization iterative algorithm, a detailed design algorithm is proposed. Finally, an example is given to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper investigates the set stability of probabilistic Boolean networks (PBNs) with switching topology. To deal with this problem, two novel concepts, set reachability and the largest invariant set family, are defined. By constructing an auxiliary system, the necessary and sufficient conditions for verifying set reachability are given and the calculation method for the largest invariant set family is obtained. Based on these two results, an equivalent condition of set stability is derived, which can be used to determine whether a PBN with switching topology can be stabilized to a given set. In addition, the design method of switching signal is proposed by combining the characteristic of the largest invariant set family, and a numerical example is reported to demonstrate the efficiency of presented approach. [ABSTRACT FROM AUTHOR]

This study discusses the robust stability problem of Boolean networks (BNs) with data loss and disturbances, where data loss is appropriately described by random Bernoulli distribution sequences. Firstly, a BN with data loss and disturbances is converted into an algebraic form via the semi-tensor product (STP) technique. Accordingly, the original system is constructed as a probabilistic augmented system, based on which the problem of stability with probability one for the original system becomes a set stability with probability one for the augmented system. Subsequently, certain criteria are proposed for the robust stability of the systems. Moreover, an algorithm is developed to verify the robust set stability of the augmented system based on truth matrices. Finally, the validity of the obtained results is demonstrated by an illustrative example. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper, we investigate observer‐based switching‐like adaptive‐triggered resilient state feedback control for discrete singular systems under deception‐injection (DI) attacks. Considering state variables are not completely measurable, a delayed state observer is engineered to reconstruct system states and the occurrence of DI attacks is described by Bernoulli random variables. The upper bound method is used to deal with the DI attacks with jumping patterns and imprecise occurrence probabilities. Taking into account the sporadic nature of DI attacks, switching‐like adaptive‐triggered resilient control method is presented, and the collaboratively designed triggered mechanism and resilient control strategy can not only automatically adjust data transmission based on the system states, but also save communication resources and network bandwidth, and enhance the tolerance to hostile attacks so as to improve safety and reliability of networked singular systems. By utilizing singular value decomposition technique, the noncausal behavior of the singular system is avoided and by applying linear matrix inequality (LMI) technology, the desired gains of controller and observer are achieved concurrently, then neoteric conditions about H∞$$ {H}_{\infty } $$ stochastic admissibility of closed‐loop discrete singular systems are provided. Ultimately, the validity of the proposed resilient coordination control approach is verified via a direct current (DC) motor model. [ABSTRACT FROM AUTHOR]

This paper mainly establishes the network-based evolution model for a class of hybrid dynamic games (HDG) systems by considering the fixed event-triggering mechanism. First, a general state-space model for HDG is introduced, and a seven-tuple representation for its normal form is established. Then we establish the directed network models, which include a tree network structure form and a multi-layer temporal model. Such network models present the advantages of evolution analysis on HDG. Finally, we use HDG example with the Lanchester equation to demonstrate the feasibility of the model and the reliability of its evolutionary process. [ABSTRACT FROM AUTHOR]

• The set stabilization problem of Markovian jump Boolean control networks by semi tensor product is studied. • A necessary and sufficient criterion for the feedback set stabilization of Markovian jump Boolean control networks is proposed based on average optimal control problem. • A policy iterati algorithm for set stabilization problem is deduced, based on which, the state feedback control is designed. In this paper, the authors address the set stabilization problem for Markovian jump Boolean control networks (MJBCNs) by state feedback control. Based on the semi-tensor product, a necessary and sufficient condition for the set stabilization of MJBCNs is put forward via optimal control approach. Moreover, a policy iterative algorithm is deduced for designing the state feedback control. Finally, the proposed method and results are supported by the example of an apoptosis network. [ABSTRACT FROM AUTHOR]

Stabilization and set stabilization of switched Boolean control networks are investigated by using flipping mechanism in this paper. Firstly, with the help of Warshall algorithm, an explicit criterion for the stabilization of switched Boolean control networks is derived. Secondly, the necessary and sufficient condition for the solvability of stabilization of switched Boolean control networks, by flipping some elements of perturbation set once, is presented. Thirdly, a search algorithm is proposed to calculate the minimum number of stabilization flipped nodes and what exactly they are. Furthermore, a necessary and sufficient condition is established for the solvability of set stabilization of switched Boolean control networks by flipping some elements of perturbation set once. Analogously, an algorithm is given to find the minimum number of set stabilization flipped nodes. Finally, examples are shown to demonstrate the feasibility of the above results. • An explicit criterion for the stabilization of switched Boolean control networks is derived. • Necessary and sufficient condition for the stabilization of switched Boolean control networks, by flipping some elements of perturbation set once, is presented. • A search algorithm is proposed to calculate the minimum number of stabilization flipped nodes and what exactly they are. • A necessary and sufficient condition is established for the set stabilization of switched Boolean control networks by flipping some elements of perturbation set once. • An algorithm is given to find the minimum number of set stabilization flipped nodes. [ABSTRACT FROM AUTHOR]

This paper concerns the detectability problem of Boolean networks with disturbance inputs using the semi-tensor product approach. Detectability like observability describes the ability of the system state to be indirectly measured, which is an important issue to be considered in designing the controller. Four kinds of detectability, weak, weak periodical, strong and strong periodical ones, are defined. A systematic matrix method is developed and formed via three tools respectively called detection operator, detection matrix and detectable set. The former is used to generate the state estimate at each time. The other two are derived from the dynamics of the system and produce several necessary and sufficient conditions for checking detectability. Two examples are provided to demonstrate the effectiveness of the main results. Furthermore, how the proposed methods can be applied and extended to other kinds of disturbance is briefly discussed. [ABSTRACT FROM AUTHOR]

• Compared with the existing results, an improved algorithm which can effectively reduce computational complexity of calculating the maximum control invariant set is given. • An explicit criterion is derived to check whether a switched Boolean network satisfies set stabilization. • An event-triggered mechanism is formulated in the process of designing time-varying set stabilization state feedback controller. • A systematic procedure is devised to design event-triggered set stabilization state feedback controller of switched Boolean networks. • A method is proposed to obtain all possible event-triggered state feedback controllers for set stabilization of switched Boolean networks. In this paper, the set stabilization problem of switched Boolean networks is reinvestigated. An event-triggered mechanism is formulated in the process of designing time-varying set stabilization state feedback controller. First, compared with the existing results, an improved algorithm which can effectively reduce computational complexity of calculating the maximum control invariant set is given. Second, using Warshall algorithm, an explicit criterion is derived to check whether a switched Boolean network satisfies set stabilization. Third, a systematic procedure is devised to design event-triggered set stabilization state feedback controller of switched Boolean networks. Then, a method is proposed to obtain all possible event-triggered state feedback controllers for set stabilization of switched Boolean networks. Finally, two examples are shown to demonstrate the effectiveness of presented results. [ABSTRACT FROM AUTHOR]

This paper introduces the detectability notions and investigates the detectability properties for Boolean networks (BNs) and Boolean control networks (BCNs). A BN (BCN) possesses the property of detectability if the current state can be determined by limited observed data. According to different situations, three types of detectabilities are defined: strong detectability, detectability and weak detectability. Firstly, using the semi-tensor product of matrices, the BCN is written as an equivalent algebraic expression, which is called the data form. Secondly, some criterion and detection algorithms are provided for checking detectabilities via this novel form. Thirdly, four types of properties: stability, stabilizability, observability and detectability, are compared and analyzed, and then their relationships are clearly described in two diagrams, which provide a new perspective to analyze the characterization of BNs and BCNs. Finally, for better presentation, two practical examples are presented to show the effectiveness of the obtained new results. [ABSTRACT FROM AUTHOR]

This paper investigates the reconstructibility of switched Boolean control networks (SBCNs). Several new types of definition about reconstructibility are proposed where the existence and arbitrariness of the switching signal and the input sequence are inconsistent. A weighted pair graph describing all pairs of states which are indistinguished is defined. A deterministic finite automata method is applied into the reconstructibility analysis. Accordingly, several algorithms are designed to determine these types of reconstructibility. Moreover, for a given reconstructible SBCN, an algorithm for determining the current state is provided. Finally, an example is provided to demonstrate the effectiveness of the proposed algorithms and results. [ABSTRACT FROM AUTHOR]

This paper considers the set controllability of Boolean control networks with impulsive effects (BCNs-IE). To investigate this problem, set controllability matrices are proposed via semi-tensor product of matrices. As applications, two control problems, which are the controllability of BCNs-IE with mix-type controls and the output controllability of BCNs-IE, are solved by set controllability approach. [ABSTRACT FROM AUTHOR]

This paper investigates zero-determinant (ZD) strategies for iterated multi-player multi-action games with multiple memories. First, using the semi-tensor product (STP) of matrices, an equivalent algebraic model for the iterated game is proposed. Based on the algebraic expression, ZD strategies are derived for repeated finite games with multiple memories, allowing players to unilaterally apply a linear relationship among their augmented expected payoffs regardless of their opponents' strategies. Furthermore, ZD strategies for repeated symmetric and skew-symmetric games are considered, based on which the coupling correlations among players' payoffs, as well as the structure of the payoff vectors, are clarified. It is found that in repeated symmetric games, any player can get the same augmented expected payoff by employing a ZD strategy. In repeated skew-symmetric games, it is possible for all participants to achieve a zero-sum outcome. To execute the aforementioned ZD strategies, some identical elements are required in each augmented payoff vector for symmetric games, whereas the payoff vector for skew-symmetric games must contain zero elements. [ABSTRACT FROM AUTHOR]

In this paper, two kinds of stabilizability of switched Boolean networks (SBNs), consistent stabilizability and stabilizability under time-varying output feedback (TVOF), are investigated on the basis of adjoint logical networks (ALNs), which are constructed via switching signals. Firstly, globally consistent stabilizability of SBNs is equivalently converted into local (one state) pointwise stabilizability of ALNs, under which all suitable finite-time switching signals can be designed. Then the method is generalized to design stabilizable TVOF switching signals. Finally, to show the effectiveness of obtained results, two types of stabilizability of a shift register are discussed. [ABSTRACT FROM AUTHOR]

This paper studies the output consensus problem of leader–follower multiagent systems (MASs) over finite fields via semi-tensor product (STP). First, the leader–follower MASs over finite fields are modeled as algebraic form, based on which, the output consensus problem is transformed into the set stabilization problem about the systems with algebraic form. Second, all the output feedback control invariant sets (OFCISs) are calculated from the state feedback maximum control invariant set (SFMCIS). Third, some criteria of the output consensus as well as the design method of output feedback controller are presented by the truth matrix technique. Finally, an example is shown to demonstrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

• The structure matrices for most finite Boolean-type lattices and complementation algebras are presented. • Using the structure matrices, the homomorphisms and isomorphisms of BTAs are investigated. • The decomposition of a BTA is considered, which means decomposing a BTA into a product of two BTAs, and a straightforward verifiable necessary and sufficient condition is obtained. This paper provides a systematic matrix description for finite Boolean- type algebras (BTAs). A BTA is decomposed into Boolean-type lattice (BTL) and a complementation algebra (CA). When the object set is finite, the matrix expressions of BTL and CA (and then of BTA) are presented. Via their matrix expression, the construction and certain properties of BTAs are investigated, including the homomorphism and isomorphism, as well as the decomposition. Moreover, straightforward verifiable conditions are obtained to detect the properties above by means of logical matrices operations. [ABSTRACT FROM AUTHOR]

This paper proposes some new sufficient conditions and necessary conditions for exponential mean stability with prescribed decay rate, and sufficient conditions for non-fragile controller design of both continuous-time and discrete-time positive Markov jump linear systems with time-delay. First, sufficient stability conditions are derived by constructing novel linear stochastic co-positive Lyapunov-Krasovskii functionals. Second, the corresponding necessary conditions are established by applying a model transformation technique and analyzing the relationship between stochastic stability of the transformed systems and exponential mean stability with given decay rate of the original systems. Compared with the existing conditions, the proposed stability conditions are not only delay-dependent but also decay-rate-dependent, and the sufficient conditions plus the necessary conditions can be used to judge the system stability more precisely. Third, based on the new sufficient conditions, non-fragile controllers are designed by solving linear programming problems such that the closed-loop systems are positive and exponentially mean stable with an expected decay rate. Finally, by numerical examples, the effects of both time-delay and decay rate on exponential mean stability are exploited and the validity of the non-fragile controller design conditions is demonstrated. [ABSTRACT FROM AUTHOR]

This article proposes a dimensionality reduction approach to study the output regulation problem (ORP) of Boolean control networks (BCNs), which has much lower computational complexity than previous results. First, an auxiliary system which is much smaller in scale than the augmented system in previous approach is constructed. By analyzing the set stabilization of the auxiliary system as well as the original BCN, a necessary and sufficient condition to detect the solvability of the ORP is presented. Second, a method to design the state feedback controls for the ORP is proposed. Finally, two biological examples are given to demonstrate the effectiveness and advantage of the obtained new results.

This paper investigates the identification of time-delay Boolean networks (TBNs) and time-delay Boolean control networks (TBCNs) via Cheng product. According to all admissible (input-)output sequences, definition on identifiability of the (TBCN) TBN is given. Two algorithms are designed to select suitable delay parameters of the TBN and TBCN, respectively. Based on these, the original systems are divided into several subsystems. Then by virtue of observability, the criteria for identifiability of the TBN and TBCN are obtained. Moreover, the corresponding constructing processes are presented to establish the internal structures of the TBN and TBCN. Finally, two illustrative examples are given to show the feasibility of the proposed methods., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 ISA. Published by Elsevier Ltd. All rights reserved.)

In this paper, the reachability of dimension-bounded linear systems is investigated. Since state dimensions of dimension-bounded linear systems vary with time, the expression of state dimension at each time is provided. A method for judging the reachability of a given vector space $ \mathcal{V}_{r} $ is proposed. In addition, this paper proves that the $ t $-step reachable subset is a linear space, and gives a computing method. The $ t $-step reachability of a given state is verified via a rank condition. Furthermore, annihilator polynomials are discussed and employed to illustrate the relationship between the invariant space and the reachable subset after the invariant time point $ t^{\ast} $. The inclusion relation between reachable subsets at times $ t^{\ast}+i $ and $ t^{\ast}+j $ is shown via an example.