Dmitry Khlopin
The extension of a conflict control problem with infinite horizon is constructed. This extension is the projective limit of restricted games. Relations between "sensitivity to target set" and the existence of the optimal control are studied. Special attention is paid to the pursuit-evasion game with "joint control/relaxed control".
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We study a pursuit-evasion differential game with finite number of pursuers and one evader in Hilbert space with geometric constraints on the control functions of players. We solve the game by presenting explicit strategies for pursuers which guarantee their pursuit as well as an strategy for the evader which guarantees its evasion.
Safe Control for Pursuit-Evasion with Density Functions
针对追逃问题，提出基于密度函数的安全控制框架，将追逃微分博弈转化为鲁棒安全控制问题，通过凸平方和规划求解，避免了Hamilton-Jacobi-Isaacs方程的高复杂度，仿真验证了方法有效性。
A qualitative account is given of a differential pursuit/evasion game. A criterion for the existence of an intercept solution is obtained using future cones that contain all attainable trajectories of target or interceptor originating from an initial position. A sufficient and necessary conditon that an opportunity to intercept always exist is that, after some initial time, the future cone of the target be contained within the future cone of the interceptor. The sufficient condition may be regarded as a kind of Nash equillibrium.
Pursuit and evasion conflicts represent challenging problems with important applications in aerospace and robotics. In pursuit-evasion problems, synthesis of intelligent actions must consider the adversary's potential strategies. Differential game theory provides an adequate framework to analyze possible outcomes of the conflict without assuming particular behaviors by the opponent. This article presents an organized introduction of pursuit-evasion differential games with an overview of recent advances in the area. First, a summary of the seminal work is outlined, highlighting important contri…
Analytical Pursuit-Evasion Game Strategy in Arbitrary Keplerian Reference Orbits
本文提出了一种适用于任意开普勒参考轨道的分析性追捕-逃避博弈策略，通过解析解实现高效求解，显著提升计算效率并适用于复杂轨道场景。
A Numerical Analysis for Pursuit-Evasion Games under the Stackelberg Eqilibrium
本文提出了一种在Stackelberg均衡下求解追捕-躲避微分博弈的数值分析框架，引入Semi-DCNLP方法用于飞行路径优化，并验证了该方法的有效性。
New strategies for some pursuit-evasion differential games with Gronwall-type constraints
本文在序列空间$l_{2}$中研究受Gronwall型约束的非合作追逃微分博弈，构造追方新策略、逃方容许策略，建立对应充分条件，扩展了微分博弈相关现有结果。
We consider a pursuit-evasion differential game with a Hybrid system of dynamics in Hilbert space with integral constraints on the control functions of players. We show that the pursuer has a winning strategy.
This paper investigates obstacle-free simple motion pursuit-evasion problems where the pursuer is faster and game termination is point capture. It is well known that the interior of the Apollonius Circle (AC) is the evader's dominance region, however, it was unclear whether the evader could reach outside the initial AC without being captured. We construct a pursuit strategy that guarantees the capture of an evader within an arbitrarily close neighborhood of the initial AC. The pursuer strategy is derived by reformulating the game into a nonlinear control problem, and the guarantee holds agains…
Here we present a decomposition technique for a class of differential games. The technique consists in a decomposition of the target set which produces, for geometrical reasons, a decomposition in the dimensionality of the problem. Using some elements of Hamilton-Jacobi equations theory, we find a relation between the regularity of the solution and the possibility to decompose the problem. We use this technique to solve a pursuit evasion game with multiple agents.
Solution and Optimal Strategies for a Differential Game of Pursuit-Evasion with Point-Wise Constraints
研究序列空间$l_2$中受逐点约束、含可数追逃者与单个逃避者的微分博弈，明确双方动力学规则与收益目标，给出博弈解、最优策略构造及博弈值估计。
A geometric approach to differential game theory is illustrated. The parallel pursuit is considered as a two-player zero-sum differential game. The optimal strategies of each player is designed based on Riemann-Finsler geometry. Our approach incorporates a closed loop optimal control and the presentation is familiar with a Brachistochrone type problem.
Analysis of First-Order Linear Pursuit and Evasion Differential Games with Gronwall-Type Constraints
该论文研究n维空间中受格朗沃尔型约束的一阶线性追逃微分对策，分析追击结果与参与者可达域的关系，并给出逃避可行的充分条件。
A single player and a mass of agents: a pursuit evasion-like game
本研究探讨单个玩家与大量智能体间的有限时域追逃微分博弈，利用输运方程描述群体动态，推导无限维Isaacs方程，并证明值函数为希尔伯特空间不变子集上的唯一粘性解。
A qualitative criterion for a pursuer to intercept a target in a class of differential games is obtained in terms of \emph{future cones}: Topological cones that contain all attainable trajectories of target or interceptor originating from an initial position. An interception solution exists after some initial time iff the future cone of the target lies within the future cone of the interceptor. The solution may be regarded as a kind of Nash equillibrium. This result is applied to two examples: 1. The game of Two Cars: The future cone condition is shown to be equivalent to conditions for interc…
Elliptical Pursuit and Evasion -Extended Version-
本研究针对一对一椭圆追逃问题，验证了追方轨迹形状仅与双方速度比有关、不受逃方速度变化影响的假设，并补充分析了不同轨道下追逃动力系统的特性及相关微分方程。
In a pursuit-evasion game, a team of pursuers attempt to capture an evader. The players alternate turns, move with equal speed, and have full information about the state of the game. We consider the most restictive capture condition: a pursuer must become colocated with the evader to win the game. We prove two general results about pursuit-evasion games in topological spaces. First, we show that one pursuer has a winning strategy in any CAT(0) space under this restrictive capture criterion. This complements a result of Alexander, Bishop and Ghrist, who provide a winning strategy for a game wit…
Pursuit-Evasion on a Sphere and When It Can Be Considered Flat
本文研究球面上的追逃博弈，将平面阿波罗尼斯圆推广为球面阿波罗尼斯域，推导出拦截点属于该域的条件，并据此分析了两追捕者对抗一较慢逃逸者的策略且进行了数值模拟。
In this paper, we address a pursuit-evasion game involving multiple players by utilizing tools and techniques from reinforcement learning and matrix game theory. In particular, we consider the problem of steering an evader to a goal destination while avoiding capture by multiple pursuers, which is a high-dimensional and computationally intractable problem in general. In our proposed approach, we first formulate the multi-agent pursuit-evasion game as a sequence of discrete matrix games. Next, in order to simplify the solution process, we transform the high-dimensional state space into a low-di…
Dynamical Systems in Elliptical Pursuit and Evasion
本文研究一对一追击逃避问题中圆形与椭圆形的差异，通过假设追击者轨迹形状不受逃避者速度影响，推导出动力系统，并分析了捕获时间上界和周期解的存在性。
Dominance Regions of Pursuit-evasion Games in Non-anticipative Information Patterns
本文研究非预期信息模式下追逃博弈中逃避者支配区域的性质，首次证明初始支配区域等于开环可达区域，并给出无障碍、有障碍及单角点障碍场景下追捕者非预期捕获策略存在的条件。
This paper studies a two-person linear-quadratic-Gaussian pursuit-evasion differential game with costly but controlled information. One player can decide when to observe the other player's state. However, one observation of another player's state comes with two costs: the direct cost of observing and the implicit cost of exposing his state. We call games of this type a Pursuit-Evasion-Exposure-Concealment (PEEC) game. The PEEC game constitutes two types of strategies: The control strategies and the observation strategies. We fully characterize the Nash control strategies of the PEEC game using…
Value of Multi-pursuer Single-evader Pursuit-evasion Game with Terminal Cost of Evader's Position: Relaxation of Convexity Condition
本研究提出了一种无需凸性条件的追捕游戏价值函数计算方法，通过几何方法和粘性解理论，解决了终端成本局部Lipschitz连续的非凸情况。
Cooperative Nonlinear Guidance Strategies for Guaranteed Pursuit-Evasion
本文研究三体追逃问题，为逃跑者-防御者团队提出基于几何的协同非线性制导策略，保证在任意初始交战几何下拦截追击者，且无需获知对方策略，误差变量可在预定有限时间内收敛至零。
This paper considers a pursuit-evasion scenario among three agents -- an evader, a pursuer, and a defender. We design cooperative guidance laws for the evader and the defender team to safeguard the evader from an attacking pursuer. Unlike differential games, optimal control formulations, and other heuristic methods, we propose a novel perspective on designing effective nonlinear feedback control laws for the evader-defender team using a time-constrained guidance approach. The evader lures the pursuer on the collision course by offering itself as bait. At the same time, the defender protects th…
Two-player pursuit-evasion differential game and time optimal zero control problem in $\ell^2$ are considered. Optimal control for the corresponding zero control problem is found. A strategy for the pursuer that guarantees the solution for the pursuit problem is constructed.
We formulate and analyze game-theoretic problems for systems governed by integral equations. For Volterra integral equations, we obtain and prove necessary and sufficient conditions for linear-quadratic problems, and for problems that are linear-quadratic in the control. Also, we obtain necessary conditions for one type of pursuit-evasion Volterra games.
Pursuit-evasion differential games using the Disturbance Attenuation approach are revisited. Under this approach, the pursuer actions are considered to be control actions, whereas all external actions, such as target maneuvers, measurement errors and initial position uncertainties, are considered to be disturbances. Two open issues have been addressed, namely the effect of noise on the control gains, and the effect of trajectory shaping on the solution. These issues are closely related to the question of the best choice for the disturbance attenuation ratio. Detailed analyses are performed for…
We study computability-theoretic aspects of differential games. Our focus is on pursuit and evasion games played in Euclidean spaces in the tradition of Rado's "Lion versus Man" game. In some ways, these games can be viewed as continuous versions of reachability games. We prove basic undecidability of differential games, and study natural classes of pursuit-evasion games in Euclidean spaces where the winners can win via computable strategies. The winning strategy for Man found by Besicovitch for the traditional "Lion versus Man" is not computable. We show how to modify it to yield a computable…
In this paper, we investigate a pursuit-evasion game in which a mobile observer tries to track a target in an environment containing obstacles. We formulate the game as an optimal control problem with state inequality constraint in a simple environment. We show that for some initial conditions, there are two different regimes in the optimal strategy of the pursuer depending on whether the state-constraint is activated. We derive the equations that characterize the switching time between the two regimes. The pursuer's optimal tracking strategy in a simple environment is further extended to a ge…
Convex Pursuit-Evasion Games for Spacecraft Proximity Operations on Circular and Elliptical Orbits
针对航天器近距离操作的追击-逃逸博弈，提出基于支撑函数的逃逸证书与安全策略，提供精确单侧保证，并利用投影外梯度法快速求解，适用于圆轨道与椭圆轨道。
The pursuit-evasion game with two persons is considered. Both players are moving in a metric space, have equal maximum speeds and complete information about the location of each other. We study the sufficient conditions for a capture (with a positive capture radius). We assume that Lion wins if he manages the capture independently of the initial positions of the players and the evader's strategy. We prove that the discrete-time simple pursuit strategy is a Lion's winning strategy in a compact geodesic space satisfying the betweenness property. In particular, it means that Lion wins in compact …
Interception Conditions in Multi-Agent Pursuit Games Governed by Linear Dynamics with Gronwall-type Constraints
本文针对受Gronwall型约束的线性动力学多智能体追逃问题，构建追捕者许可策略并推导拦截保证的充分条件，通过数值示例验证策略有效性与分析条件。
A defender-attacker-target problem with non-moving target is considered. This problem is modeled by a pursuit-evasion zero-sum differential game with linear dynamics and quadratic cost functional. In this game the pursuer is the defender, while the evader is the attacker. The objective of the pursuer is to minimize the cost functional, while the evader has two objectives: to maximize the cost functional and to keep a given terminal state inequality constraint. The open-loop saddle point solution of this game is obtained in the case where the transfer functions of the controllers for the defend…
This paper studies a planar multiplayer Homicidal Chauffeur reach-avoid differential game, where each pursuer is a Dubins car and each evader has simple motion. The pursuers aim to protect a goal region cooperatively from the evaders. Due to the high-dimensional strategy space among pursuers, we decompose the whole game into multiple one-pursuer-one-evader subgames, each of which is solved in an analytical approach instead of solving Hamilton-Jacobi-Isaacs equations. For each subgame, an evasion region (ER) is introduced, based on which a pursuit strategy guaranteeing the winning of a simple-m…
Compact metric spaces with infinite cop number
本文通过在$\mathbb{S}^3$上构造具有无限警察数的度量，反驳了Mohar关于紧致测地度量空间中有限警察必胜及同调群秩上界的猜想。
In this paper, we address pursuit-evasion problems involving multiple pursuers and multiple evaders. The pursuer and the evader teams are assumed to be heterogeneous, in the sense that each team has agents with different speed capabilities. The pursuers are all assumed to be following a constant bearing strategy. A dynamic divide and conquer approach, where at every time instant each evader is assigned to a set of pursuers based on the instantaneous positions of all the players, is introduced to solve the multi-agent pursuit problem. In this regard, the corresponding multi-pursuer single-evade…
Dynamic zero-sum games are an important class of problems with applications ranging from evasion-pursuit and heads-up poker to certain adversarial versions of control problems such as multi-armed bandit and multiclass queuing problems. These games are generally very difficult to solve even when one player's strategy is fixed, and so constructing and evaluating good sub-optimal policies for each player is an important practical problem. In this paper, we propose the use of information relaxations to construct dual lower and upper bounds on the optimal value of the game. We note that the informa…
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