This article investigates a multileader Stackelberg game where leaders lack critical information about the follower's objective function and face random disturbances with unknown distributions. Unlike conventional approaches requiring complete follower information, we consider leaders who manipulate physical plant states while observing the follower's strategy through private tracking responders. To address distributional uncertainty in the follower's best response, we reformulate the game as a distributionally robust equilibrium-seeking problem and develop a fully distributed feedback learning algorithm. The proposed data-driven approach operates without prior knowledge of system models or disturbance distributions, enabling leaders to estimate states through neighbor communication and local gradient updates. We characterize equilibrium existence in nonconvex settings. The relationship between communication and gradient errors and the energy function of the dynamic system is established. The upper bound of the regret based on the proposed algorithm is rigorously analyzed. A case study demonstrates the framework's effectiveness in achieving distributionally robust solutions against uncertain stochastic perturbations.

Distributed optimization, as a key technology for collaborative intelligence in multiagent systems, has been widely applied in sensor networks, deep learning, and smart grids. Although numerous effective algorithms have been proposed, classical methods typically rely on idealized assumptions, such as accurate objective information, perfect communication channels, and trustworthy system environments. However, these assumptions are frequently violated in real-world applications. To bridge the gap between theory and practice, distributed optimization under information constraints has emerged as a research focus. This survey provides a systematic overview of recent advances in this field. We categorize information constraints based on their origin into three primary types: i) observational constraints, including stochastic objectives, online optimization, and zeroth-order methods; ii) communication constraints, such as random network topologies, delays, asynchronous updates, and communication-efficient strategies; and iii) system-level constraints, encompassing privacy preservation and Byzantine-resilient optimization. This survey reviews the research progress and challenges associated with each constraint category. Furthermore, we use two representative case studies to analyze the practical application of these algorithms and the origins of information constraints in real-world problems. Finally, we explore promising future research directions.

This article presents a novel model-free distributionally robust framework for a challenging equilibrium-seeking problem (ESP) under fully unknown coupled dynamics. We consider a scenario in the ESP where the state transitions of players are governed by an unknown coupled dynamic system, and each player aims to minimize its own cost function. By predicting the stochastic distribution of player states through Gaussian process regression, we propose a novel distributionally robust approximation (DRA) that transforms the complex ESP with unknown coupled dynamic system into a solvable distributionally robust optimization problem. The gradient of the DRA's objective function is quantified, ensuring solvability. The effectiveness of the proposed DRA framework is evaluated through a nonlinear system, demonstrating comparable performance to model-based methods without requiring any dynamic model.

Game theory has emerged as a fundamental framework for modeling and analyzing strategic interactions and decision-making among multiple agents, and has witnessed rapidly growing impact in cyber–physical systems over the past decade. Its integration with dynamic systems has driven major theoretical and technological advances in a wide range of applications, including smart grids, autonomous driving, robotic swarms, and networked control systems. In particular, distributed games in dynamic systems and their equilibrium learning mechanisms have attracted increasing attention due to their scalability, lightweight information exchange, and real-time implementability. This article provides a comprehensive survey of distributed games in dynamic systems, where agents interact only with local neighbors while collectively achieving global equilibrium and stability. First, the foundational theories of distributed dynamic games under three representative classes of systems: linear dynamic systems, nonlinear dynamic systems, and uncertain dynamic systems, are presented. Then, state-of-the-art distributed equilibrium learning and control methods are reviewed, including gradient-based dynamics, payoff-based learning, best-response dynamics, and learning-based approaches. To demonstrate the practical relevance and impact of distributed games in dynamic systems, representative application domains are discussed in detail. Finally, several promising future research directions are outlined, highlighting open challenges at the intersection of distributed games, learning, and dynamic systems.

Digital task-oriented semantic communication (ToSC) aims to transmit only task-relevant information, significantly reducing communication overhead. Existing ToSC methods typically rely on learned codebooks to encode semantic features and map them to constellation symbols. However, these codebooks are often sparsely activated, resulting in low spectral efficiency and underutilization of channel capacity. This highlights a key challenge: how to design a codebook that not only supports task-specific inference but also approaches the theoretical limits of channel capacity. To address this challenge, we construct a spectral efficiency-aware codebook design framework that explicitly incorporates the codebook activation probability into the optimization process. Beyond maximizing task performance, we introduce the Wasserstein (WS) distance as a regularization metric to minimize the gap between the learned activation distribution and the optimal channel input distribution. Furthermore, we reinterpret WS theory from a generative perspective to align with the semantic nature of ToSC. Combining the above two aspects, we propose a WS-based adaptive hybrid distribution scheme, termed WS-DC, which learns compact, task-driven and channel-aware latent representations. Experimental results demonstrate that WS-DC not only outperforms existing approaches in inference accuracy but also significantly improves codebook efficiency, offering a promising direction toward capacity-approaching semantic communication systems.

In this article, we investigate the problem of distributed fault detection for a class of cyber-physical system whose physical layer consists of numerous subsystems, each modeled as a linear discrete-time system. Considering the influence of process noise and measurement noise, the state estimation of each subsystem is completed using a distributed Kalman filter (DKF), in which the one-step prediction is corrected not only by the local innovation but also by the measurement errors of the neighbors at the previous step. Leveraging the DKF, a local residual generator is designed for each subsystem. The parameters of the DKF are then determined by minimizing the estimation error and the upper bound of its covariance in the fault-free case, which ensures the robustness of the residual. Furthermore, by utilizing the instantaneous $T^{2}$ test statistic and the sliding window-based $T^{2}$ test statistic of the residual signals, the corresponding residual evaluation function and fault detection threshold are established to facilitate fault detection for each subsystem. In the proposed fault detection scheme, each subsystem only transmits information to its neighbors, ensuring that each subsystem can detect its faults in a distributed manner. Additionally, a sufficient condition is provided to guarantee the mean square boundedness of the estimation error in the fault-free case. Finally, a power network system is employed to demonstrate the effectiveness of the proposed scheme.

This article investigates a noncooperative game of multiagent systems in incomplete information scenarios. To cooperatively seek the Nash equilibrium (NE), each agent aims to minimize its own cost function by interacting with its neighbors over undirected communication networks. While existing distributed NE seeking methods alleviate the computational burden, they also entail higher communication costs. To reduce communication frequency and bandwidth, we propose a class of distributed edge-based NE seeking methods by leveraging the advantages of event-triggered mechanisms and quantization techniques. In the proposed framework, a buffer is equipped on every communication channel, thereby reducing the workload of both agents at either end. It is shown that the convergence error can be made arbitrarily small by tuning a constant threshold, and it can asymptotically converge to zero by setting an exponentially decaying threshold or a dynamic threshold. Moreover, in the case of unawareness of any global information, we further provide a fully distributed event-triggered quantized algorithm, by which the convergence error is ultimately uniformly bounded. Finally, two numerical examples are utilized to illustrate the effectiveness of the proposed algorithms.

This paper proposes a zero-gradient-sum-based time-varying distributed prescribed-time optimization algorithm for single integrator dynamics. The algorithm introduces a prescribed-time sliding mode term aimed at attaining zero-gradient-sum, alongside a sliding-mode controller designed to ensure consensus among agents' states within a prescribed-time limit. Notably, the algorithm eliminates the need for initial condition constraints and local minimization. The criteria for achieving consensus and optimizing multi-agent systems are derived from optimization theory and Lyapunov stability theory. Finally, the superior convergence efficiency of the algorithm is verified through a power-sharing case study.

In the past few years, energy management strategies based on multi-agent reinforcement learning (MARL) have been an active research topic. However, existing MARL algorithms require a massive amount of data, making it challenging to ensure data privacy. To address the problem, this paper investigates FL-MARL that combines federated learning (FL) with MARL. FL allows each agent to train based on local data and only share model parameters, which means that the actual data are not shared among agents. The framework is in conjunction with a MARL algorithm: independent proximal policy optimization (IPPO). The proposed algorithm has two advantages: 1) It can protect data privacy of each agent and 2) It can adapt to large-scale and decentralized data scenarios. Finally, the performance of the algorithm is verified by simulation.

This paper investigates a distributed prescribed-time observer design for a type of multi-agent systems under homologous sensor attacks. First, we design Luenberger-like observers based on attack consensus which can accurately obtain the state of the system at prescribed time. Then, the sufficiency condition and construction algorithm of the observer gain matrix design are given. Compared with the existing results of distributed state estimation under homologous sensor attacks, the observer proposed in this paper does not need to use all data in the time window and the prescribed convergence time is independent of the initial value of the system. Simulation results validate the usefulness of the designed observer.

This article investigates the optimal bipartite consensus control (OBCC) problem for unknown second-order discrete-time multiagent systems (MASs). First, the coopetition network is constructed to describe the cooperative and competitive relationships between agents, and the OBCC problem is proposed by the tracking error and related performance index function. Based on the distributed policy gradient reinforcement learning (RL) theory, a data-driven distributed optimal control strategy is obtained to guarantee the bipartite consensus of all agents' position and velocity states. In addition, the offline data sets ensure the learning efficiency of the system. These data sets are generated by running the system in real time. Besides, the designed algorithm is an asynchronous version, which is essential to solve the challenge caused by the computational ability difference between nodes in MASs. Then, by means of the functional analysis and Lyapunov theory, the stability of the proposed MASs and the convergence of the learning process are analyzed. Furthermore, an actor-critic structure containing two neural networks is used to implement the proposed methods. Finally, a numerical simulation shows the effectiveness and validity of the results.

In this article, an observer-based adaptive neural network (NN) event-triggered distributed consensus tracking problem is investigated for nonlinear multiagent systems with quantization. In the first place, the limited capacity of the communication channel between agents is considered. The event-trigger mechanism and dynamic uniform quantizers are set up to reduce information transmission. The next NN is utilized to handle the unknown nonlinear functions. Finally, in order to estimate the unmeasurable states, an NN-based state observer is designed for each agent by using a dynamic gain function. To settle the difficulty caused by the coupling effects of event-triggered conditions and the scaling function in dynamic uniform quantizers and observers, a distributed control protocol with estimated information of its neighbors is designed, which ensures distributed consensus tracking of the nonlinear multiagent systems without incurring the Zeno behavior. The effectiveness of the control protocol is illustrated by a simulation example.

In this article, we shall investigate the event-triggered communication control problem for strict-feedback nonlinear systems with measurement outputs. First, two event-triggered communication schemes are designed. Based on both event-triggered schemes, the measurement output and control input signals are only transmitted at triggering time instants, which saves communication costs from the sensor to the controller and from the controller to the actuator. Meanwhile, Zeno behavior can be excluded under the proposed triggering schemes. Second, since the full-state information is not available to the controller, by developing an observer, the system state is estimated and a controller based on estimated state information is designed. Due to the irregular sampling of information communication and state estimation error affects each other, the parameters of the state observer, the controller, and the event-triggering mechanism should be jointly designed. It is proved that the closed-loop system state converges to the origin. Finally, a simulation example verifies the validity of the obtained theoretical result.

This article proposes a dynamic gain approach for investigating the leader-follower consensus problem of nonlinear multiagent systems with both distributed delays and discrete delays. It is assumed that each agent is in the strict-feedback form and satisfies the Lipschitz condition with an unknown constant. A novel dynamic gain observer is first constructed for each follower by only utilizing the output information of the follower and its neighbors. By introducing an observer, a distributed output-feedback control protocol is designed for each follower agent under a directed communication topology. It is proved by the Lyapunov-Razumikhin theorem that the consensus of the multiagent system is achieved with the proposed consensus protocol. Different from the existing results, the dynamic gain is constructed for each follower to propose the consensus protocol, without resorting to the celebrated backstepping technique. Moreover, the time-varying gain is designed in a uniform framework. The effectiveness of the proposed approaches is illustrated by a simulation example.

This article aims to address the problem of distributed energy management for both the generation and demand sides in smart grid. Different from many existing works, we investigate the SWM problem with transmission losses. In addition, instead of transforming the local constraint into an approximate penalty function or the projection set, we consider it as a convex nonsmooth indicator function from a different viewpoint. For such a composite problem consisting of smooth and nonsmooth terms, we propose a distributed proximal primal–dual algorithm based on dual decomposition and operator splitting techniques. Each node performs the algorithm through only local computation and communication with limited information, especially not sharing the sensitive gradient directly. It is also proved that the proposed algorithm leads to the global optima at a convergence rate$O(\frac{1}{k})$with a fixed stepsize. Several simulations verify the theoretical analysis and demonstrate the effectiveness of the proposed algorithm.

In this paper, we consider the control problem of multi-agent systems with additive noises in the uncertain situation, (which means we are not sure about a particular probability space.) With mild condition, both G-mean square convergence and G-mean square steady-state error are analyzed by graph theory and G-martingale convergence. Different from classical probability, weak convergence in capacity and weak convergence in capacity 1 are considered.

This paper is concerned with solving a large category of convex optimization problems using a group of agents, each only being accessible to its individual convex cost function. The optimization problems are modeled as minimizing the sum of all the agents' cost functions. The communication process between agents is described by a sequence of time-varying yet balanced directed graphs which are assumed to be uniformly strongly connected. Taking into account the fact that the communication channel bandwidth is limited, for each agent we introduce a vector-valued quantizer with finite quantization levels to preprocess the information to be exchanged. We exploit an event-triggered broadcasting technique to guide information exchange, further reducing the communication cost of the network. By jointly designing the dynamic event-triggered encoding-decoding schemes and the event-triggered sampling rules (to analytically determine the sampling time instant sequence for each agent), a distributed subgradient descent algorithm with constrained information exchange is proposed. By selecting the appropriate quantization levels, all the agents' states asymptotically converge to a consensus value which is also the optimal solution to the optimization problem, without committing saturation of all the quantizers. We find that one bit of information exchange across each connected channel can guarantee that the optimiztion problem can be exactly solved. Theoretical analysis shows that the event-triggered subgradient descent algorithm with constrained data rate of networks converges at the rate of O(lnt√t). We supply a numerical simulation experiment to demonstrate the effectiveness of the proposed algorithm and to validate the correctness of theoretical results.

We study a distributed constrained optimal consensus problem of discrete-time multi-agent systems under fixed communication delays. Specifically, the total cost is expressed as the sum of individual cost of each agent, and only part of agents have access to the constraint. The constrained optimal consensus is solved if the final consensus value falls within the constraint, and in the meanwhile minimizes the total cost. Based on consensus method and subgradients, we propose a distributed two-step update scheme in which the state of each agent is firstly averaged with the delayed state information from neighbors, followed by a decaying subgradient descent from the individual cost, together with a movement along the projection direction if the agent can access the constraint. We show that the distributed constrained optimal consensus problem under fixed communication delays can be solved if the fixed network is balanced and contains a spanning tree, the constraint is accessible by at least one agent, and the gain on subgradient is decaying but persistent. Simulation results are provided to verify our conclusion.

In this paper, we consider the distributed parameter estimation problem over sensor networks in the presence of quantized data and directed communication links. We propose a two-stage algorithm aiming at achieving the centralized sample mean estimate in a distributed manner. The running average technique is utilized in the proposed algorithm to smear out the randomness caused by the probabilistic quantization scheme. It is shown that the centralized estimate can be achieved in the mean square sense, which is not observed in the conventional consensus algorithms. Simulation results are presented to illustrate the effectiveness of the proposed algorithm and highlight the improvements by using running average technique.

This paper considers the average consensus problem for multi-agent systems with continuous-time first-order dynamics. The communication channels among the agents are constrained in which the exchanged information is quantized. In this paper, logarithmic quantization is considered in the communication channels, and sampled-data based protocol is applied. It is shown that as long as the sampling interval is small enough, the consensus protocol is admissible under arbitrarily coarse quantization. To be specific, the consensus error is uniformly bounded and is proportional to the quantization error and averaged initial value. Numerical examples are given to demonstrate the effectiveness of the protocol.

In this paper, we consider formation control problem for multi-robot systems under an undirected communication network. All the robots will track a leader, while form a desired formation. The leader can be static or dynamic. A distributed formation controller with neighbors' input information is applied. For practical implementation, control input information from neighbors can only be received after some time delays. It is therefore shown that the distributed control protocol using time-delayed control input information from neighbors guarantees the formation of the multi-agent system for any nonnegative delay. We will implement the new protocol on the Amigo robots. The experimental results will demonstrate the effectiveness of the new protocol.