Differential Game Research Articles

We investigate the interception problem in a differential game with non-inertial players (a pursuer and an evader) who move in dynamic flow fields with various influences. Throughout the paper, we solve the pursuit and ``life-line'' game problems. To solve the pursuit, the strategy of parallel pursuit ($\bf{\Pi}$-strategy for short) is defined and used. With the help of the $\bf{\Pi}$-strategy and applying the Gr\"{o}nwall-Bellman inequality, sufficient pursuit condition is determined. In order to solve the ``life-line'' game to the advantage of the pursuer, we build the set of meeting points of the players and prove that this set monotonically decreases with regard to inclusion relative to time. The ``life-line'' game to the advantage of the evader is solved by constructing evader's attainability domain where it reaches without being caught for an arbitrary control of the pursuer.

We analyze an evasion differential game involving one evader and multiple pursuers in $\R^n$. The control functions of the players are subject to exponential integral constraints to ensure bounded energy consumption. Evasion is considered possible if, for any time $t$, the position of the evader differs from the positions of all the pursuers. In this work, we establish a sufficient condition for the possibility of evasion. We construct an admissible evasion strategy and demonstrate that, for any number of pursuers $m$, evasion is possible. Additionally, we show that the number of maneuvers required for evasion does not exceed $m$.

Sustainable strategies for marine plastic waste remanufacturing systems under diverse carbon reduction policies.

Blockchain-enabled governance of greenwashing in the building material supply chain: A differential game approach

Target defense differential game for autonomous surface vehicles

Non-zero-sum linear quadratic stochastic differential games with jump diffusion and input delay: an asymmetric information framework

Abstract The title of this semi-tutorial, expository paper might be applicable to the remarkable body of research by Professor Tamer Başar with his students Didinski and Pan in the 1990s on robustification of parameter identifiers and adaptive controllers through game-theoretic methods. The author’s inspiration indeed fuses their work with his results from that period on inverse optimal adaptive stabilization and inverse optimal disturbance attenuation in a differential game formulation. Inverse optimal adaptive and minimax techniques are merged in this paper’s first half to obtain solutions to adaptive differential games between the adaptive controller and a disturbance. The article’s second half introduces a recent advance in disturbance-robustification of adaptive control for persistent disturbances (merely bounded, rather than square-integrable), by Iasson Karafyllis and this article’s author. While not with a minimax capability, this robustification is first in forty years to attain regulation with a bias that is arbitrarily low and independent of both the unknown parameter and the persistent disturbance.

Parameter estimation for stochastic dynamic systems is crucial yet challenging due to inherent randomness and noise. Traditional deterministic methods often fail to cope with such uncertainties. Inspired by the deterministic tracking approach, we propose a new parameter estimation method—stochastic dynamic problem based parameter estimation method (SDPE), which firstly transforms the original stochastic optimal control problem into a stochastic differential game problem by introducing a perturbation term, and then realizes the simultaneous estimation of the system parameters and the optimal control function. We prove the consistency of the SDPE estimator. Numerical simulations demonstrate that this method exhibits superior performance in terms of accuracy and robustness compared to maximum likelihood estimation (MLE), likelihood-based methods (EM) and Bayesian inference methods (MCMC), particularly under varying noise levels and sample sizes. Furthermore, we apply SDPE to a stochastic optimal control SIRD model to analyze COVID-19 data from Florida, USA. Results demonstrate that SDPE effectively captures time-varying transmission, recovery, and mortality rates, providing a powerful tool for modeling and understanding complex epidemic dynamics.

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

A differential game analysis of firms' Greentech innovation with emission tax in oligopoly exhibiting network externality

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.