This article focuses on prescribed-time distributed robust Nash equilibrium seeking for monotone games impacted by unknown and time-varying disturbances. First, a regularization term with a prescribed-time decaying parameter is introduced to compensate for the absence of strong monotonicity in the merely monotone game. Based on the regularization technique, a new prescribed-time signum-based distributed Nash equilibrium seeking algorithm incorporating an integral sliding mode method, a leader-following consensus protocol, and a gradient algorithm is presented for monotone games with unknown but bounded disturbances. Then, to dispose of the unknown bounds of disturbances, a prescribed-time distributed adaptive integral sliding mode based Nash equilibrium seeking strategy is devised. On the basis of the proposed strategies, some sufficient conditions are obtained to guarantee that the players' actions are capable of converging to the least-norm Nash equilibrium of the monotone games in a prescribed time. In the end, numerical simulations on least-distance formation control of a network of players testify to the performance of the proposed seeking strategies.

Distributed control of networked Euler–Lagrange systems has broad applications in industrial engineering. However, challenges, such as multiple uncertainties and resource-constrained networks, remain urgent issues to be resolved. This article addresses the issue of distributed Nash equilibrium seeking for Euler–Lagrange systems subject to physical and cyber uncertainties and limited communication resources. Specifically, considering uncertain parameters and time-varying uncertainties imposed on communication links, a new distributed strategy integrating an optimizer, a state regulator, a consensus algorithm, and an adaptive law is proposed for Euler–Lagrange systems, in which gains for consensus modules adaptively adjust to cope with the compromised communication weights caused by cyber uncertainties. Moreover, a dynamic gradient-based event-triggered scheme is put forward to enable information exchanges among neighbors and control updates when the predetermined triggered conditions are met. It is shown by theoretical analysis that the proposed event-triggered strategy is effective for significantly reducing the numbers of information transmission and control updates nearly without degrading convergence performance. Furthermore, the Zeno phenomenon is theoretically precluded under the proposed event-triggered scheme. Finally, the efficacy of the proposed method is validated through simulation cases on robotic manipulators.

This paper considers a network of distributed energy resources (DERs) that intend to optimize their power generation in an open price-based energy management system. Specifically, in the considered problem, the cost function of each DER, depends on not only its own decision variable but also an aggregate of all DERs’ energy decision variables. The DERs are cooperative to minimize their total cost of power generations, thus forming a distributed aggregative optimization problem. In addition, it is considered that the DERs are allowed to frequently arrive in and leave the system, which brings great difficulties in the algorithm design and analysis. To address this problem, a novel distributed optimization algorithm is proposed, by integrating gradient descent algorithms with a multi-level storage-based consensus mechanism. In addition, the performance of the proposed algorithm is evaluated by the dynamic regret, which quantifies the sum of all DERs’ loss between the actual cost and real time optimal cost during its active period length. It is shown that the upper bound of the all active DERs’ dynamic regret over active period length grows sublinearly under diminishing stepsizes. Finally, a numerical simulation is provided to verify the effectiveness of the proposed method.

This work studies the problem of finding optimal distributed resource allocation policies on wireless networks in the existence of an unknown malicious adding-edge attacker, depicted as the games of games (GoG) model. Specifically, two subnetwork policymakers constitute a Nash game, while a Stackelberg game captures the confrontation between each subnetwork policymaker and the malicious attacker. We first demonstrate that by utilizing the Foschini-Miljanic algorithm, the communication resource allocation of cellular networks can be converted into a geometric program (GP) that can be efficiently solved by convex optimization. Then, the upper limit of attack magnitude that the network can withstand is calculated. It is shown that the proposed GP framework is solvable within the attack bound and a Gestalt Nash equilibrium (GNE) exists for the GoG. Moreover, a heuristic algorithm that iteratively employs GP is developed to obtain the optimal policy profiles of subnetworks, which can converge asymptotically. Correspondingly, a greedy heuristic adding-edge strategy is proposed to identify the set of the most vulnerable edges for the attacker. Finally, simulation examples show that the obtained algorithm is resilient to malicious attacks and can attain the GNE. Despite the presence of malicious attacks, all channels’ transmission and interference gains can be well-adjusted within a limited budget.

This article is concerned with fully distributed Nash equilibrium seeking in networked games under both undirected and directed communication graphs. New fully Nash equilibrium seeking strategies incorporating gradient-based optimization algorithms, consensus algorithms, and double-layer adaptive control laws are presented. In particular, the double-layer adaptive control laws are introduced to ensure that the control gains are not overlarge and free of dependence on any global information. This is achieved by adding a damping term to the adaptive parameter design such that the continuous increase in control gains is avoided. Theoretical analyses are conducted to prove that players' actions can be convergent to the Nash equilibrium under the proposed strategies. Moreover, it is shown that the developed strategies can be extended to accommodate the players with heterogeneous linear dynamics. Finally, numerical examples are provided to illustrate the effectiveness of the proposed methods.

This paper studies attack-defense differential games for multiple 3D defenders and 2D or 3D attackers, where the attackers aim to attack a target region protected by the defenders with non-negative capture radii. Since the direct analysis of many-to-many games is difficult, the problem is decomposed into multiple many-to-one subgames. For the subgame between multiple 3D defenders and a 2D attacker, this paper presents a concept of attack region and constructs a defense winning strategy based on the attack region. Based on this strategy, it is proved that if a defense coalition can win, there are no more than two members that can guarantee this winning. The analysis is extended to 3D attackers with a lowest flight altitude constraint. Under this defense strategy, the defenders move towards an estimated interception point, and the optimality of the strategy is shown. The bipartite matching with constraints is adopted to resolve many-to-many games based on subgame outcomes. Simulations are provided to validate the theoretical results.

This paper considers resilient Nash equilibrium seeking for linear dynamic players under false data injection (FDI) attacks. Different from most of existing works that consider secure networks, players' actuators and communication paths are both compromised by malicious attackers in this paper. A distributed algorithm is constructed for the players based on state regulators, consensus protocols, false data observers and gradient algorithms. The proposed algorithm guarantees that players' actions can asymptotically converge to the Nash equilibrium under both actuator and communication channel attacks. At last, numerical examples are provided to demonstrate the effectiveness of the proposed resilient seeking algorithm.

Nash equilibrium, as an essential strategic profile in game theory, is of both practical relevance and theoretical significance due to its wide penetration into various fields, such as smart grids, wireless communication networks, and networked mobile vehicles. In particular, distributed Nash equilibrium seeking strategies have recently attracted increasing attention because they show remarkable advantages in relaxing the requirement of a central node for information broadcasting or full observation of players’ actions. This article aims to provide a survey of distributed Nash equilibrium seeking in games with partial decision information, in which players can only exchange information with their neighbors and their objective functions may explicitly depend on all players’ actions. First, fundamental problem descriptions on distributed Nash equilibrium seeking are presented. Second, related results on distributed Nash equilibrium seeking in general multiplayer games, aggregative games, and multicluster games are reviewed, respectively, where representative continuous- and discrete-time methods are explained in detail. Third, two practical applications, including collaborative control for a network of mobile sensors and energy consumption control in smart grids, are provided to demonstrate the applicability of distributed Nash equilibrium seeking strategies. Finally, some promising directions are suggested for future research.

In practical applications, decision makers with heterogeneous dynamics may be engaged in the same decision-making process. This motivates us to study distributed Nash equilibrium seeking for games in which players are mixed-order (first- and second-order) integrators influenced by unknown dynamics and external disturbances in this article. To solve this problem, we employ an adaptive neural network to manage unknown dynamics and disturbances, based on which a distributed Nash equilibrium seeking algorithm is developed by further adapting concepts from gradient-based optimization and multiagent consensus. By constructing appropriate Lyapunov functions, we analytically prove the convergence of the reported method. Theoretical investigations suggest that players’ actions would be steered to an arbitrarily small neighborhood of the Nash equilibrium, which is also testified by simulations.

This paper is concerned with distributed Nash equilibrium seeking problems involving double-integrator-type players who do not have knowledge on their own velocity signals. Moreover, the control inputs of players are considered to be affected by time-varying disturbances. To solve this problem, an uncertainty and disturbance estimator (UDE) is adopted to construct a distributed robust Nash equilibrium seeker. By our theoretical analysis, it is found that players' actions can be driven to be arbitrarily close to the Nash equilibrium by utilizing the developed method. In addition, for constant and exponentially decaying disturbances, asymptotic results can be derived. Finally, a simulation example is provided to illustrate the presented theoretical results.

This article aims to accommodate networked games in which the players' dynamics are subjected to unmodeled and disturbance terms. The unmodeled and disturbance terms are regarded as extended states for which observers are designed to estimate them. Compensating the players' dynamics with the observed values, the control laws are designed to achieve the robust seeking of the Nash equilibrium for networked games. First, we consider the case in which the players' dynamics are subject to time-varying disturbances only. In this case, the seeking strategy is developed by employing a smooth observer based on the proportional-integral (PI) control. By utilizing the designed strategy, we show that the players' actions would converge to a small neighborhood of the Nash equilibrium. Moreover, the ultimate bound can be adjusted to be arbitrarily small by tuning the control gains. Then, we further consider the case in which both an unmodeled term and a disturbance term coexist in the players' dynamics. In this case, we adapt the idea from the robust integral of the sign of the error (RISE) method in the strategy design to achieve the asymptotic seeking of the Nash equilibrium. Both strategies are analytically investigated via the Lyapunov stability analysis. The applications of the proposed methods for a network of velocity-actuated vehicles are discussed. Finally, the effectiveness of the proposed methods is verified via conducting numerical simulations.

The functionality of energy routing among microgrids is becoming increasingly important with the progress of deploying smart power systems all over the world. For higher energy routing performance and better renewable energy integration, a new type of electrical device, called energy router (ER), is being developed as a part of the infrastructure of the future energy Internet (EI). Generally, the long-term operation of ERs requires an effective energy management scheme for the energy storage inside these devices. In this article, considering the randomness of power generation by renewable energy sources and the stochastic power usage of loads in EI scenario, the compressive sensing is adopted for the solution to the nonlinear energy storage management problem which is essential for the design of ERs. The compressive sensing method used in this article is proven to be more efficient than the conventional Monte–Carlo methods and polynomial chaos expansion method, and the performance of the proposed method is evaluated with numerical examples.

With the incentive to solve Nash equilibrium computation problems for networked games, this article tries to find answers for the following two problems: 1) how to accommodate hybrid games, which contain both continuous-time players and discrete-time players? and 2) are there any other potential perspectives for solving continuous-time networked games except for the consensus-based gradient-like algorithm established in our previous works? With these two problems in mind, the study of this article leads to the following results: 1) a hybrid gradient search algorithm and a consensus-based hybrid gradient-like algorithm are proposed for hybrid games with their convergence results analytically investigated. In the proposed hybrid strategies, continuous-time players adopt continuous-time algorithms for action updating, while discrete-time players update their actions at each sampling time instant and 2) based on the idea of consensus tracking, the Nash equilibrium learning problem for continuous-time games is reformulated and two new computation strategies are subsequently established. Finally, the proposed strategies are numerically validated.

This article studies distributed optimization algorithms for heterogeneous multiagent systems under an undirected and connected communication graph. Two types of heterogeneities are discussed. First, we consider a class of multiagent systems composed of both continuous-time dynamic agents and discrete-time dynamic agents. The agents coordinate with each other to minimize a global objective function that is the sum of their local convex objective functions. A distributed subgradient method is proposed for each agent in the network. It is proved that driven by the proposed updating law, the agents' position states converge to an optimal solution of the optimization problem, provided that the subgradients of the objective functions are bounded, the step size is not summable but square summable, and the sampling period is bounded by some constant. Second, we consider a class of multiagent systems composed of both first-order dynamic agents and second-order dynamic agents. It is proved that the agents' position states converge to the unique optimal solution if the objective functions are strongly convex, continuously differentiable, and the gradients are globally Lipschitz. Numerical examples are given to verify the conclusions.

A social cost minimization problem is addressed in this article. In the considered problem, a network of agents work collaboratively to minimize the social cost function, which is defined as the sum of the agents’ local objective functions. The engaged agents are supposed to be equipped with an undirected and connected communication graph. Different from most of the existing works, the social cost function in the considered problem is allowed to be nonconvex and possibly admits local extrema. To avoid local extrema and achieve the global minimization of the social cost function, an extremum-seeking-based approach is proposed by introducing a dynamic average consensus protocol to the sinusoidal-dither-signal-based extremum seeking scheme. The dynamic average consensus protocol is leveraged in the proposed extremum-seeker for information sharing and the sinusoidal probing signal is utilized for information extraction. For the avoidance of local extrema, the amplitude of the dither signal is designed to be adaptive. Through Lyapunov stability analysis, it is shown that the proposed method enables the decision variable to converge to a neighborhood of the global minimum point if the conditions on the network connectivity, the existence of unique global minimum and achievability of the global minimum are satisfied. The theoretical result is verified via simulating a numerical example.

Nash equilibrium seeking for games in a class of hybrid systems is investigated in this paper. Different from the existing works, the players in the present game framework are composite of a set of continuous-time players and a set of discrete-time players. Nash equilibrium seeking for such a hybrid game is challenging as some players update their actions in continuous time while the remainders update their actions in discrete time. To accommodate the hybrid games, we firstly consider a case in which the players update their actions according to the hybrid gradient play (i.e., the continuous-time players update their actions according to the continuous-time gradient play with sampled information flow while the discrete-time players update their actions according to the discrete-time gradient play). Then, we consider the case in which the players have restricted access to their opponents' actions. A hybrid consensus-based strategy is proposed for this case. The stability of the Nash equilibrium under the proposed hybrid seeking strategies is theoretically proven by utilizing Lyapunov stability analysis. Lastly, the hybrid seeking strategies are validated through a numerical example.

This paper investigates distributed Nash equilibrium seeking in multiagent games under switching communication topologies. To be specific, the communication topology is supposed to be switching among a set of strongly connected digraphs, which might suffer from occasional loss of communication due to sensor failure, packet loss, etc. The synthesis of the leader-following consensus protocol and the gradient play is exploited to achieve the distributed Nash equilibrium seeking under the switching communication topologies. Switching topology without loss of communication is firstly considered, followed by switching topology subject to missing communication within some time slots. For both situations, nonquadratic and quadratic games are addressed separately. Local convergence results are presented for nonquadratic games and nonlocal convergence results are provided for quadratic games. The theoretical results are verified by numerical examples.

In this paper, an aggregate game is adopted for the modeling and analysis of energy consumption control in smart grid. Since the electricity users' cost functions depend on the aggregate energy consumption, which is unknown to the end users, an average consensus protocol is employed to estimate it. By neighboring communication among the users about their estimations on the aggregate energy consumption, Nash seeking strategies are developed. Convergence properties are explored for the proposed Nash seeking strategies. For energy consumption game that may have multiple isolated Nash equilibria, a local convergence result is derived. The convergence is established by utilizing singular perturbation analysis and Lyapunov stability analysis. Energy consumption control for a network of heating, ventilation, and air conditioning systems is investigated. Based on the uniqueness of the Nash equilibrium, it is shown that the players' actions converge to a neighborhood of the unique Nash equilibrium nonlocally. More specially, if the unique Nash equilibrium is an inner Nash equilibrium, an exponential convergence result is obtained. Energy consumption game with stubborn players is studied. In this case, the actions of the rational players can be driven to a neighborhood of their best response strategies by using the proposed method. Numerical examples are presented to verify the effectiveness of the proposed methods.

We solve N -player potential games with dynamical constraint in this paper. Potential games with stable dynamics are first considered followed by one type of potential games without inherently stable dynamics. Different from most of the existing Nash seeking methods, we provide an extremum seeking-based method that does not require explicit information on the game dynamics or the payoff functions. Only measurements of the payoff functions are needed in the game strategy synthesis. Lie bracket approximation is used for the analysis of the proposed Nash seeking scheme. A singularly semi-globally practically uniformly asymptotically stable result is presented for potential games with stable dynamics and an ultimately bounded result is provided for potential games without inherently stable dynamics. For first-order perturbed integrator-type dynamics, we employ an extended-state observer to deal with the disturbance such that better convergence is achievable. Stability of the closed-loop system is proven and the ultimate bound is quantified. Numerical examples are presented to verify the effectiveness of the proposed methods.