Abstract This study proposes a geometric solution to the norm differential game design problem in target-attacker-defender (TAD) engagements, addressing key limitations of conventional zero-effort-miss approaches. By leveraging the geometric analogy between guidance-law-generated trajectories and Dubins paths, we reformulate the derivation of zero-effort-miss-based guidance laws as a Nash equilibrium optimisation problem, with optimal strategies determined through reachable set analysis of Dubins path frontier. The resulting model is a non-convex optimisation problem, which prevents the derivation of traditional state-feedback control laws. To overcome this limitation and enable real-time implementation, we develop a custom back propagation neural network, enhanced with a relaxation factor method for output filtering, a Holt linear trend model for outlier compensation and a saturation function for oscillation suppression. Extensive simulations demonstrate that the proposed framework significantly outperforms baseline methods. These results validate the effectiveness and robustness of our approach for high-performance TAD applications.

This paper delves into a multi-player non-renewable resource extraction differential game model, where the duration of the game is a random variable with a composite distribution function. We first explore the conditions under which the cooperative solution also constitutes a Nash equilibrium, thereby extending the theoretical framework from a fixed duration to the more complex and realistic setting of random duration. Assuming that players are unaware of the switching moment of the distribution function, we derive optimal estimates in both time-dependent and state-dependent cases. The findings contribute to a deeper understanding of strategic decision-making in resource extraction under uncertainty and have implications for various fields where random durations and cooperative strategies are relevant.

Abstract We analyze the tradeoff between mitigation and adaptation in response to irreversible climate disaster in a transboundary pollution differential game with a stochastic regime shift. Countries can reduce their emissions to decrease the likelihood that the shift occurs, or proactively invest in adaptation to reduce the impact of the shift. We characterize the social optimum and the non-cooperative Markov-perfect Nash equilibrium (MPNE). In the social optimum, adaptation complements mitigation and increases expected welfare. In contrast, in the MPNE the option to adapt crowds out mitigation, reduces precautionary behavior, and decreases expected welfare. Our analysis stresses the importance of strategic interactions when countries can use multiple policy instruments to combat climate change.

This paper is concerned with a three-level multi-leader-follower incentive Stackelberg game with $H_\infty$ constraint. Based on $H_2/H_\infty$ control theory, we firstly obtain the worst-case disturbance and the team-optimal strategy by dealing with a nonzero-sum stochastic differential game. The main objective is to establish an incentive Stackelberg strategy set of the three-level hierarchy in which the whole system achieves the top leader's team-optimal solution and attenuates the external disturbance under $H_\infty$ constraint. On the other hand, followers on the bottom two levels in turn attain their state feedback Nash equilibrium, ensuring incentive Stackelberg strategies while considering the worst-case disturbance. By convex analysis theory, maximum principle and decoupling technique, the three-level incentive Stackelberg strategy set is obtained. Finally, a numerical example is given to illustrate the existence of the proposed strategy set.

An approximate optimal control issue for modular unmanned systems (MUSs) is presented via a cooperative differential game for solving the trajectory tracking problem. Initially, the modular unmanned system’s dynamic model is built with the joint torque feedback technique. The moment of inertia of the motor rotor has positive symmetry. Each MUS module is deemed as a participant in the cooperative differential game. Then, the MUS trajectory tracking problem is transformed into an approximate optimal control problem by means of adaptive critic design (ACD). The approximate optimal control is obtained by the critic network, approaching the joint performance index function of the system. The stability of the closed-loop system is proved through Lyapunov theory. The feasibility of the proposed control algorithm is verified by an experimental platform.

In this paper, we construct a supply chain coordination model that considers the complex relationships among dynamic pricing, advertising investment, and consumer welfare. The model focuses on a supply chain consisting of a manufacturer and a retailer, where the manufacturer produces the product and the retailer sells it. We first analyze the equilibrium under decentralized decision-making, in which the retailer is the leader and the manufacturer the follower. By building a differential game model, we derive the optimal pricing and advertising strategies under decentralization. Then, we set a centralized optimal benchmark in which the supply chain acts as an integrated decision maker. Comparing decentralized and centralized decisions, we find a double-marginalization effect under decentralization that reduces the total profit of the supply chain. To address this, we design a series of coordinating contracts—including revenue sharing, price discount, and two-part tariff contracts—among which revenue sharing can effectively coordinate the chain and maximize total profit. We further analyze the effect of advertising on consumer welfare. Using a consumer welfare measurement model, we find that advertising increases consumer surplus and thus improves consumer welfare. Numerical simulations verify the effectiveness of the model and the superiority of the coordinating contracts.

Finite-time differential games in missile guidance: Achieving motion camouflage and robust cooperative encirclement guidance.

Cooperative Differential Game Guidance Laws for Target With Multiple Defenders Against Missile

Differential Games for a Mixed ODE-PDE System

Abstract In the practice of Greentech innovation, there are two distinct paths: the first is replacement technology innovation, aimed at improving replacement technology, and the second is end‐of‐pipe technology innovation, aimed at reducing end‐of‐pipe emissions. This paper examines Greentech innovation and emission taxation in an oligopoly market with network externality. Its key features include (i) extending the literature on classical Greentech innovation by modeling it as a dynamic oligopoly with network externality; (ii) incorporating an inverse demand function that depends on output levels and network value; and (iii) imposing emission taxes on emissions rather than environmental damage. The main findings are (i) a saddle‐point steady‐state equilibrium exists under both firm and social planner decisions; (ii) an increase in efforts for replacement technology innovation leads to higher output and prices, whereas an increase in efforts for end‐of‐pipe technology innovation leads to lower output and higher prices; (iii) emissions increase (decrease) with increased efforts for replacement technology innovation when the network size is large (small), while emissions consistently decrease with increased efforts for end‐of‐pipe technology innovation, regardless of network size; (iv) in the steady‐state equilibrium, the social planner can achieve full Greentech adoption by either setting an appropriate tax rate (for a given number of firms) or regulating market access (for a given tax rate). Additional insights are presented through propositions and conclusions. This study complements and advances the classical economies of scale approach to Greentech innovation by shifting the focus from production‐side to demand‐side dynamics.

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.

Game theory based vision impedance control for human-robot interaction.

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