The finite multi-objective potential game (FMOPG) with different objective sets is proposed in this paper. Leveraging the semi-tensor product of matrices, two necessary and sufficient conditions for verifying FMOPGs have been presented, which provide the theoretical foundation for the research on potential-based multi-objective distributed optimization. In addition, the Pareto equilibrium in this class of games is examined by establishing two equivalent verification criteria. Significantly, the relationship between Pareto equilibrium and potential functions is further characterized, which demonstrates that this model can exhibit richer equilibrium behaviors and model more complex interaction patterns compared to FMOPGs with a common objective set for all players. Finally, taking multi-objective facility-based systems as an example, the applicability of the results is validated.

In scenarios such as voting and elections, the endeavor to achieve a certain degree of consensus is widespread. Consequently, in this paper, the approximate consensus problem of networked evolutionary matrix games is proposed to better describe such situations. First, the verification of (locally) approximate consensus is given from both algebraic and topological perspectives. Secondly, taking the consumption for operators and the compensation for puppet players into account, a more reasonable and cost-effective finite-cost event-triggered mechanism, which is composed of non-functional control and asynchronous control by different operators, is proposed to facilitate approximate consensus. Under this module, a necessary and sufficient condition is derived to verify the existence of this mechanism. Then, the shortest time and minimum cost for achieving approximate consensus, as well as the corresponding event-triggered mechanisms, are obtained by breadth-first search algorithms. Finally, the conclusions are extended to the scenarios including groups and unfixed approximate consensus and the theoretical result is demonstrated effectively by an illustrative example.

We propose a novel fictitious-play-based learning algorithm, called group-based joint strategy fictitious play (GJSFP) with inertia, to seek group-based Nash equilibria of group-based generalized ordinal potential games. The proposed GJSFP with inertia enables group-based Nash equilibria to possess an “ab sorption” property, and allows group-based generalized ordinal potential games to converge to group-based Nash equilibria under somecertain conditions. Notably, the convergence of group-based generalized ordinal potential games without additional conditions can be guaranteed under a fading memory GJSFP with inertia. For group-based networked potential games, fading memory GJSFP with inertia ensures that such games converge to a profile that is both a group-based Nash equilibrium and an individual-based Nash equilibrium.

This paper addresses the distributed multi-objective optimization problem for discrete-time heterogeneous multi-agent systems via potential games. Potential game-based methods are widely employed in distributed optimization for multi-agent systems, as they enable the decoupling of solution processes and ensure convergence to the desirable equilibrium. While potential game theory is effective for single-objective cases, its multi-objective extension lacks a systematic framework, is limited by strict assumptions, and exhibits poor explainability. To overcome these limitations, a novel semi-tensor product (STP)-based framework is proposed, which is one of the powerful tools for the research of finite potential games. The main contributions are (1) formulating a novel game model-finite multi-objective networked potential games (MONPGs)-for heterogeneous interactions, with an STP-based algebraic condition enabling local information-based payoff design; (2) designing a strategy learning algorithm that guarantees the convergence to a Pareto equilibrium and is universally applicable to arbitrary real-valued payoff vectors, significantly enhancing generality compared to prior works; and (3) deriving a sufficient condition expressed as a linear matrix equation for solving the distributed optimization problem. This work extends potential game-based methods to multi-objective and heterogeneous interaction settings, enhancing interpretability and solving a class of problems previously intractable for existing potential game-based methods.

In this paper, the stabilization problem of linear time-invariant (LTI) systems over finite fields is investigated based on the fully-actuated system (FAS) approach. Unlike the existing studies that primarily focus on naturally FASs, this paper addresses general LTI systems over finite fields, including sub-actuated, over-actuated, and under-actuated cases. By employing rank decomposition and controllability structure decomposition over finite fields, it is shown that any LTI system can be equivalently transformed into a FAS or a cascade consisting of a FAS and an autonomous subsystem. On this basis, the stabilization problem is reformulated based on the FAS approach. It is further proved that the original system is stabilizable if and only if the associated autonomous subsystem is nilpotent. An algorithmic procedure is presented to systematically construct the equivalent FAS representation and design the control law, and the results are further applied to the tracking problem. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method.

A fuzzy skill map assigns a fuzzy set of skills to each problem, representing the level of proficiency required to solve it. A student is assumed to solve a problem if their competence meets the corresponding requirements; otherwise, the problem is regarded as unsolved. However, students whose proficiency falls short may still produce partial solutions rather than failing completely. Building on this observation, we extend the framework by identifying partial solutions under the same fuzzy skill map, thereby developing a skill proficiency model for delineating polytomous knowledge structures via L-fuzzy rough approximation operators. Unlike existing polytomous extensions that require reassigning fuzzy skill sets to each response of each problem, our approach builds directly on the original fuzzy skill map. Furthermore, by incorporating intuitionistic L-fuzzy approximation operators, we enhance the model to simultaneously capture both skill proficiency and misconceptions of fuzzy skills, where a fuzzy skill refers to a skill at a specific proficiency level. Finally, as a direct application of the skill proficiency model, we propose a method for providing personalized learning guidance that recommends which unmastered fuzzy skills should be learned next and which mastered fuzzy skills should be reinforced. We further provide illustrative examples to clarify how both the skill proficiency model and the corresponding learning guidance method can be used.

In this study, the stabilization problem of nonlinear strict-feedback systems (SFSs) over finite fields is investigated based on high-order fully actuated system (HOFAS) approaches. First, a nonlinear SFS with uniform dimensions is transformed into both a step forward and a step backward HOFAS. Then, control laws are designed for step backward HOFASs, and criteria for the asymptotic stability over finite fields of the resulting closed-loop system are established. Furthermore, the nonlinear SFS with increasing dimensions over finite fields is investigated and transformed into an HOFAS through a series of coordinate transformations. Finally, two numerical examples are provided to illustrate the stabilization of SFSs with uniform and increasing dimensions over finite fields, thereby validating the theoretical results. In addition, the proposed methods are applied to the orientation synchronization problem in camera networks with nonlinear protocols, and a synchronization control protocol is designed based on the HOFAS approach.

This paper focuses on the problem of converting underactuated Boolean control networks into fully actuated ones. First, Boolean control networks are equivalently transformed into an algebraic form based on the semi-tensor product of matrices. Subsequently, four types of perturbation methods are defined, namely single perturbation of a single function, multiple perturbations of a single function, single perturbation of multiple functions, and multiple perturbations of multiple functions. The necessary and (or) sufficient conditions are systematically derived to transform underactuated Boolean control networks into fully actuated ones under each scenario. Finally, three illustrative examples are provided to verify the effectiveness of the results obtained in this paper.

Nonsingularity is a key cryptographic property of feedback shift registers (FSRs). In this paper, the nonsingularity determination and propagation of nonlinear FSRs (NFSRs) are investigated using the semi-tensor product (STP), with the bisimulation method. By constructing the bisimulation equivalence relation, the nonsingularity criterion of an NFSR and the structural characteristics of the nonsingular NFSR are given, simply by checking whether the state transition matrix of its reduced system is an identity matrix. Furthermore, by introducing the concept of nonsingular bisimulation, this paper realizes the propagation of nonsingularity between NFSRs, and observability between nonsingular NFSRs with the same stage. Finally, this paper further extends this propagation of nonsingularity between NFSRs with different stages. The realization of the propagation can share the nonsingularity between different NFSRs, enhancing security in cryptographic systems.

This article addresses the fundamental issues of solvability and normalization in singular Boolean networks (SBNs) from a new perspective based on the admissible and normal initial state sets. It presents novel results and removes the restrictive conditions found in existing literature. First, the state transition matrix of SBNs is constructed by defining a new operator, and the admissible initial state set with the normal initial state set, of SBNs is introduced. Second, the problems of the solvability and uniqueness of the solution to a general SBN are converted into computing its admissible and normal initial state sets, which can be analytically computed using the derived formulas. Third, based on the normal initial state set, a necessary and sufficient condition for solving the normalization problem of general SBNs is established for the first time, which removes the restrictions in the existing literature. Finally, the results obtained are compared with existing literature and illustrated with examples.

The research on robust control of Boolean networks provides core technical support for the stable and reliable operation of complex discrete systems such as gene regulatory networks and power systems under disturbances. In previous studies, several kinds of controllers have been proposed to achieve the robust set stabilization of Boolean control networks with disturbance inputs (DBCNs). However, a key issue with the currently methods is that the controls usually update more frequently than necessary. To address this issue, this paper investigates the problem of self-triggered controller design for robust set stabilization of DBCNs. First, the definition of Lyapunov function (LF) for set stabilization of DBCNs is proposed, and an algorithm to construct the LF using truth-matrix technique is designed. Then, a necessary and sufficient condition for the set stabilization of DBCNs under the LF method is presented. Based on the obtained LF, a kind of self-triggered controllers is designed such that the system can robustly stabilize to a desired state set. Compared with existing control schemes for robust set stabilization of DBCNs, the self-triggered controllers we designed can effectively reduce the control update frequency, thereby lowering the control cost. Finally, two examples are provided to validate the efficiency of the obtained results.

This paper investigates the algebraic relationships between the dimension-keeping semi-tensor product of matrices (DK-STP) and the semi-tensor product of matrices (STP). Firstly, the definitions of five types of STPs and the type-I DK-STP are given. Secondly, the fundamental properties of the type-I DK-STP are introduced. Thirdly, the algebraic relationships between the type-I DK-STP and these five types of STPs are explored respectively. Finally, the DK-STP is generalized, and the algebraic relationships between the right DK-STP and the right STPs are discussed.

This paper applies the Cheng projection to the support vector machine (SVM) in handling missing data. In the process of handling missing data, each sample with missing values is replaced by its Cheng projection in the original space. Additionally, two classification algorithms for handling linearly separable and nonlinearly separable datasets with missing data are presented. For linearly separable datasets with missing data, Cheng kernel function is introduced, and an SVM classification algorithm that improves the linear kernel function to the Cheng kernel function is proposed. For nonlinearly separable datasets, a generalized Gaussian Radial Basis Function kernel is introduced and an SVM classification algorithm for handling missing data is given. For both algorithms, two comparative experiments are conducted to demonstrate their effectiveness.

This paper investigates the consensus problem of multi-communication networks with time-delays over finite fields. Firstly, the constant delay case is studied. Based on the algebraic topology and the graph theory, some criteria, guaranteeing that finite-field networks with multi-communication channels and constant time-delays achieve finite-field consensus, are derived. Then the inverse recursion subspaces are studied, revealing that the finite-field consensus problem can be determined by a small part of states. Along the lines of constant time-delays networks, the finite-field consensus of multi-communication networks with time-varying bounded delays is discussed. And the distinguish condition of the method using a union graph is given. Finally, a numerical example is provided to illustrate the effectiveness of the proposed results. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

It is our great honor to present this special issue of the Journal of Systems Science &Complexity in celebration of Professor Daizhan Cheng's 80th birthday.As a distinguished scholar and mentor in control theory and applied mathematics,Professor Cheng has made groundbreaking contributions that have reshaped the landscape of systems science,inspiring generations of researchers worldwide.This special issue,consisting of 18 high-quality research papers contributed by his colleagues,collaborators,and former students,reflects the breadth and depth of his intellectual influence across multiple domains,including nonlinear control,logical dynamic systems,game theory,optimization,and complex networks.

Partial occlusion tends to cause the loss of key facial features, which directly leads to a sharp decline in the performance of core tasks including facial recognition and identity verification. To address this issue, both the Compressed Sensing (CS) method and various deep learning methods have been introduced for the recognition of partially occluded facial images. Nevertheless, these methods still suffer from the drawbacks of slow computational speed and high storage overhead. In this paper, we propose a novel framework that integrates semi-tensor product compressed sensing with a dictionary matrix downsampling strategy for partially occluded face processing. This approach not only achieves the dual merits of low storage consumption and fast computation but also boasts good generalization capability, enabling rapid and accurate recognition of various occlusion types. Furthermore, extensive experimental results are presented to validate the effectiveness and practical applicability of the proposed method.

This paper investigates the set stabilization of state-based games (SBGs) via the dynamic programming theory. A novel algebraic form-based optimal control method is proposed to drive all state-based action profiles of the SBG to the recurrent state equilibrium set in minimum time. First, by introducing an optimal time vector, the set stabilization of SBGs is transformed into an optimization problem. Then, a necessary and sufficient criterion is proposed to verify the set stabilizability of SBGs. Based on this, a control algorithm that utilizes dynamic programming is presented to compute the time-optimal feedback gain matrix. For any initial state-based action profiles, the control algorithm guarantees that the profiles of all players converge to the recurrent state equilibrium set in minimal time, while maintaining low computational complexity. Finally, a numerical example validates the effectiveness of the proposed approach, demonstrating its superiority in improving computational efficiency.

Networked evolutionary games (NEGs) provide a powerful framework for modeling complex multi-agent interactions. However, existing studies often overlook the critical impact of heterogeneous communication delays, which are ubiquitous in practical networked systems. This paper investigates the infinite-horizon optimal control problem for NEGs with edge-specific delays, aiming to minimize the long-term average cost. By leveraging the semi-tensor product approach, the delayed dynamics are transformed into an equivalent algebraic representation. An augmented state profile is constructed to restore the Markov property, enabling a unified treatment of heterogeneous delays. To address the multichain structure arising from heterogeneous delays, a novel strategy iteration algorithm is developed based on the Poisson equation and a lexicographical improvement criterion. By jointly considering average and relative costs, the proposed method guarantees global optimality across disjoint invariant sets. Numerical experiments in industrial emission-reduction scenarios demonstrate that the proposed approach effectively compensates for heterogeneous delays and achieves fast convergence. The computational complexity and scalability are further analyzed, providing practical guidelines for implementation. These results establish a systematic and efficient framework for optimal control of decentralized systems with heterogeneous delays.