In recent years, with the development of micro-satellite clusters and large-scale satellite constellations, the likelihood of multiple spacecraft engaging in orbital pursuit–evasion games has increased. This paper establishes a novel interception domain theory for planar orbital multi-player “encirclement-capture” differential games, and it proves the partitioned structure and classification properties of Nash equilibrium solutions. The main contributions of our study are the following: (1) Proposing the first rigorous definition of interception domains in orbital pursuit–evasion games, proving their convexity, developing computational methods for domain intersections, and establishing a complete classification of equilibrium for planar multi-pursuer interception games, which establishes a theoretical foundation for analyzing multi-spacecraft orbital pursuit–evasion games. (2) Analyzing Nash equilibrium properties for “encirclement-capture” differential games with two, three, or more pursuers, classifying degenerate/non-degenerate scenarios via spatial inclusion relationships. The equilibrium results indicate that as the number of pursuers increases, the game tends toward a degenerate scenario where the likelihood of redundant pursuers (whose actions do not affect the game outcome) rises.

The paper provides an analysis for the behavior of equilibrium trajectories of two various types in dynamic bimatrix games, which are adequate models of evolutionary processes occurring in economic and biological systems. The model, which describes the players’ interaction, is given by differential equations with control parameters. The game dynamics has the property of strong invariance with respect to the space of the game given by unit square. Different types of game equilibria are studied for the considered dynamic model. The first type of equilibrium is obtained within the theory of differential games and based on the concept of guaranteed strategies proposed by N.N. Krasovskii. In this case, equilibrium trajectories are generated by guaranteed feedbacks and form the basic construction for the dynamic Nash equilibrium. In the second variant, equilibrium trajectories are determined on the basis of ideas of replicator dynamics from the theory of evolutionary games – a classic tool for describing competition in economic models and processes of population interaction in biology. In both cases, the designs of equilibrium trajectories are presented, for which a comparative analysis of the payoff indices is performed. A study was conducted for the trends of the trajectories of “mixed” game dynamics, within which the first player is guided by the guaranteed principles when making decisions and the second player is guided by the properties of trajectories of the replicator dynamics. The results of the comparative analysis demonstrate that the quality indices of the dynamic Nash equilibrium with guaranteed strategies dominate the characteristics of the replicator dynamics. A variant of the investment model on the financial markets of stocks and bonds is considered, which demonstrates the results of constructing equilibrium trajectories in dynamic bimatrix games.

This paper considers a state-coupled nonlinear system with constraints and disturbances. A new event-triggered distributed economic model predictive control (EMPC) strategy is proposed, which not only reduces unnecessary communication but also ensures input-to-state stability of the closed-loop system under bounded disturbances. A multi-objective optimisation approach is employed to resolve the conflict between economic performance and stability. Based on differential game theory, the economic objective function and the stability objective function concerning the optimal economic equilibrium point are optimised separately. Through specially designed contraction constraints in the EMPC optimisation problem, the recursive feasibility of the optimisation problem is ensured, and the stability of the closed-loop system under bounded disturbances is guaranteed. Additionally, a compound triggering mechanism is designed based on the deviation between actual and predicted states and the contraction constraints. Finally, the effectiveness of the proposed strategy is verified through a simulation example of a disturbed state-coupled continuous stirred-tank reactor.

Online social platform competition and rumor control in disaster scenarios: a zero-sum differential game approach with approximate dynamic programming

HighlightsWhat are the main findings?User-defined boundaries without any initial conditions are used to directly shape the tracking performance of the unknown target, which only asks for relative position and velocity sensors.Based on a skillful collision threat modeling technique called the ACSB envelope, optimal strategies and trajectories are found for each satellite, which ensures collision avoidance for the arbitrary satellite with respect to all other satellites and all specific structures of the huge target.What is the implication of the main finding?Optimal surrounding control by low-cost satellite formation of unknown huge target with non-ignorable specific structures.The issue of in-orbit optimal safe surrounding control for service satellite (SSat) formation against a huge unknown target satellite (TSat) with specific structures is solved by using relative measurements only, and an optimal cooperative safe surrounding (OCSS) hybrid controller achieving both target tracking (TT) and configuration tracking (CT) is proposed corresponding to the two equal sub-objectives. Facing the challenges caused by incomplete information of the TSat, by using relative measurements only, the initial-condition-free boundaries are constructed by an arctan-based state transformation to directly constrain the target tracking error to perform prescribed transient and steady-state behaviors. Based on the shared TT control law, optimal collision-free CT controllers for all SSats are further solved via a nonzero-sum differential game, where the collision threat from all SSats and target structures are modeled by a novel circumscribed-sphere model. Finally, the effectiveness and advantages of the proposed OCSS control technique is verified by simulation results.

Abstract We address the problem of regularity of solutions to a family of semilinear parabolic systems of equations, which describe closed‐loop equilibria of some ‐player differential games with Lagrangian having quadratic behaviour in the velocity variable, running costs and final costs . By global (semi)monotonicity assumptions on the data and , and assuming that derivatives of in directions are of order for , we prove that derivatives of enjoy the same property. The estimates are uniform in the number of players . Such a behaviour of the derivatives of arise in the theory of Mean Field Games, though here we do not make any symmetry assumption on the data. Then, by the estimates obtained we address the convergence problem in a ‘heterogeneous’ Mean Field framework, where players all observe the empirical measure of the whole population, but may react differently from one another. We also discuss some results on the joint and vanishing viscosity limit.

Finite-Agent Stochastic Differential Games on Large Graphs: I. The Linear-Quadratic Case

Multi-agent self-attention reinforcement learning for multi-USV hunting target.

Robust control in two-player nonzero-sum LQ differential games with uncertainties: A spacecraft rendezvous case study

Prescribed performance optimal fault-tolerant control for nonlinear systems with mismatched disturbances via zero-sum differential game.

Open-loop and closed-loop saddle points of infinite dimensional linear–quadratic stochastic differential games with Poisson jumps

A computational method for finding feedback Nash equilibrium solutions (FBNES) in Nonzero-Sum Differential Games (NZSDG) Based on the Variational Iteration Method (VIM)

A stochastic differential game based pollution management study of regional alliance

A jump–diffusion Stackelberg stochastic differential game in optimal carbon abatement strategies with green subsidy

A finite-horizon zero-sum linear-quadratic differential game with non-homogeneous dynamics is considered. The key feature of this game is as follows. The cost of the control of the minimizing player (the minimizer) in the game’s cost functional is much smaller than the cost of the control of the maximizing player (the maximizer) and the cost of the state variable. This smallness is due to a positive small multiplier (a small parameter) for the quadratic form of the minimizer’s control in the integrand of the cost functional. Two cases of the game’s cost functional are studied: (i) the current state cost in the integrand of the cost functional is a positive definite quadratic form; (ii) the current state cost in the integrand of the cost functional is a positive semi-definite (but non-zero) quadratic form. The latter case has not yet been considered in the literature devoted to the analysis of cheap control differential games. For each of the aforementioned cases, an asymptotic approximation (by the small parameter) of the solution to the considered game is derived. It is established that the property of the aforementioned state cost (positive definiteness/positive semi-definiteness) has an essential effect on the asymptotic analysis and solution of the differential equations (Riccati-type, linear, and trivial), appearing in the solvability conditions of the considered game. The cases (i) and (ii) require considerably different approaches to the derivation of the asymptotic solutions to these differential equations. Moreover, the case (ii) requires developing a significantly novel approach. The asymptotic solutions of the aforementioned differential equations considerably differ from each other in cases (i) and (ii). This difference yields essentially different asymptotic solutions (saddle point and value) of the considered game in these cases, meaning it is of crucial importance to distinguish cases (i) and (ii) in the study of various theoretical and real-life cheap control zero-sum linear-quadratic differential games. The asymptotic solutions of the considered game in cases (i) and (ii) are compared with each other. An academic illustrative example is presented.

Multiplayer differential games in three-dimensional (3D) space constitute a highly challenging yet profoundly significant research issue. This paper investigates a 3D reach-avoid differential game involving N pursuers vs M evaders, addressing both optimal assignments and optimal strategy simultaneously. Leveraging the Apollonius sphere as a fundamental tool, the evader’s safe region is characterized, and a representative measure for determining win–loss outcomes is proposed. Furthermore, the analytical expressions of the barrier are derived in one-on-one and two-on-one scenarios, which represent the first successful derivation of the analytical solution in 3D space. Subsequently, the optimal strategy guaranteeing the pursuer’s victory is formulated. Additionally, the basis for solving the optimal assignment is established utilizing the barrier measures. Numerical examples are given to illustrate the results. This paper lays the foundation for analyzing adversarial multiplayer differential games in high-dimensional spaces.

This paper addresses the cluster containment control problem for a swarm of linear unmanned systems operating over a directed communication graph and subject to external disturbances, formulating it as a graphical differential game. Each agent independently computes a distributed optimal control strategy using only local information, ensuring both cluster containment and Nash equilibrium behavior within the game framework, while also maintaining robustness to external disturbances. The closed-loop system is proven to be asymptotically stable, and the existence of a Nash-minmax solution for each agent is established. To implement the control strategy without requiring a model of the system dynamics, a model-free, data-driven reinforcement learning algorithm is proposed for the online computation of distributed Nash-minmax control policies, accompanied by convergence guarantees. The effectiveness of the proposed framework is demonstrated through simulations involving swarms of unmanned aerial vehicles and unmanned ground vehicles.

In the context of the Carbon Generalized System of Preferences (CGSP), this paper develops a three-tier reverse supply chain model comprising the government, recyclers, and consumers. Differential game analysis is employed to investigate the evolutionary dynamics of consumers’ perceived recycling value and to examine how government recycling efforts and recyclers’ point rewards levels influence this perception. Furthermore, the study explores the dynamic trajectory of consumers’ perceived recycling value across three decision-making models—collaborative-driven, government-driven, and market-driven—and evaluates its impact on supply chain efficiency. The research shows that (1) enhanced recycling efforts by both the government and recyclers significantly improve consumers’ perceived recycling value, thereby promoting the recycling of waste electrical and electronic equipment, with the most pronounced improvement observed under the collaborative-driven strategy; (2) in the government-driven model, the government’s subsidy rate affects recyclers’ decisions regarding point-based incentives but does not influence the government’s own recycling effort; (3) the evolutionary trajectories of consumers’ perceived recycling value and system efficiency differ among the models, with the highest levels achieved under the collaborative-driven model, followed by the government-driven model, and the lowest under the market-driven model. This study fully accounts for the dynamic nature of consumers’ perceived recycling value, offering theoretical and practical guidelines for effective WEEE recycling.

Abstract The guidance law design is considered for an engagement scenario of a missile attacking an actively defended target. A three-player nonzero-sum differential game is formulated to describe players’ respective behaviours including the attacking missile’s evading the target’s defender and thereafter attacking the target, the target’s evading the attacking missile under the assistance of the defender, and the defender’s intercepting the attacking missile to assist the target evading the attacking missile. It is considered that the missile hits the target at a specific terminal angle to enhance the destructive effect by attacking the vulnerable regions of the target. The advantages of norm-bounded and linear quadratic differential game guidance laws are combined and the cost functions segmented by relative distances between the players are proposed. The equilibrium solutions of the nonzero-sum differential game are derived and used as the respective guidance laws of the players. Simulations in various engagement scenarios are conducted to verify the reliability and robustness of the design.

In this work, an adaptive dynamic programming (ADP)-based event-triggered infinite-horizon (H∞) control algorithm is proposed for high-precision speed regulation of permanent magnet synchronous motors (PMSMs). The H∞ control problem of PMSM can be formulated as a two-player zero-sum differential game, and only a single critic neural network is needed to approximate the solution of the Hamilton–Jacobi–Isaacs (HJI) equations online, which significantly simplifies the control structure. Dynamically balancing control accuracy and update frequency through adaptive event-triggering mechanism significantly reduces the computational burden. Through theoretical analysis, the system state and critic weight estimation error are rigorously proved to be uniform ultimate boundedness, and the Zeno behavior is theoretically precluded. The simulation results verify the high accuracy tracking capability and the strong robustness of the algorithm under both load disturbance and shock load, and the event-triggering mechanism significantly reduces the computational resource consumption.