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Area-Optimal Control Strategies for Heterogeneous Multi-Agent Pursuit

Kamal Mammadov, Damith C. Ranasinghe

TL;DR

This work addresses containment in a pursuit–evasion game with heterogeneous pursuers by framing it as a zero-sum game over the area of the evader's safe-reachable set $A_e(t)$. It generalizes area-based containment from equal-speed to heterogeneous-speed pursuers and derives the area gradients $\nabla_{\mathbf{p}_i} A_e$ and $\nabla_{\mathbf{e}} A_e$ analytically via the Leibniz integral rule, without requiring a closed-form expression for $A_e(t)$. The resulting Nash-equilibrium controls are decentralized and interpretable: each active pursuer moves toward the centroid of its boundary arc, while the evader moves along a weighted sum toward arc centroids. The approach enables real-time, geometry-driven cooperation to shrink the safe region and guarantees capture in simulations, offering a principled foundation for scalable pursuit–evasion in realistic heterogeneous settings.

Abstract

This paper presents a novel strategy for a multi-agent pursuit-evasion game involving multiple faster pursuers with heterogenous speeds and a single slower evader. We define a geometric region, the evader's safe-reachable set, as the intersection of Apollonius circles derived from each pursuer-evader pair. The capture strategy is formulated as a zero-sum game where the pursuers cooperatively minimize the area of this set, while the evader seeks to maximize it, effectively playing a game of spatial containment. By deriving the analytical gradients of the safe-reachable set's area with respect to agent positions, we obtain closed-form, instantaneous optimal control laws for the heading of each agent. These strategies are computationally efficient, allowing for real-time implementation. Simulations demonstrate that the gradient-based controls effectively steer the pursuers to systematically shrink the evader's safe region, leading to guaranteed capture. This area-minimization approach provides a clear geometric objective for cooperative capture.

Area-Optimal Control Strategies for Heterogeneous Multi-Agent Pursuit

TL;DR

This work addresses containment in a pursuit–evasion game with heterogeneous pursuers by framing it as a zero-sum game over the area of the evader's safe-reachable set . It generalizes area-based containment from equal-speed to heterogeneous-speed pursuers and derives the area gradients and analytically via the Leibniz integral rule, without requiring a closed-form expression for . The resulting Nash-equilibrium controls are decentralized and interpretable: each active pursuer moves toward the centroid of its boundary arc, while the evader moves along a weighted sum toward arc centroids. The approach enables real-time, geometry-driven cooperation to shrink the safe region and guarantees capture in simulations, offering a principled foundation for scalable pursuit–evasion in realistic heterogeneous settings.

Abstract

This paper presents a novel strategy for a multi-agent pursuit-evasion game involving multiple faster pursuers with heterogenous speeds and a single slower evader. We define a geometric region, the evader's safe-reachable set, as the intersection of Apollonius circles derived from each pursuer-evader pair. The capture strategy is formulated as a zero-sum game where the pursuers cooperatively minimize the area of this set, while the evader seeks to maximize it, effectively playing a game of spatial containment. By deriving the analytical gradients of the safe-reachable set's area with respect to agent positions, we obtain closed-form, instantaneous optimal control laws for the heading of each agent. These strategies are computationally efficient, allowing for real-time implementation. Simulations demonstrate that the gradient-based controls effectively steer the pursuers to systematically shrink the evader's safe region, leading to guaranteed capture. This area-minimization approach provides a clear geometric objective for cooperative capture.

Paper Structure

This paper contains 15 sections, 4 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Area-Based Optimal Control Strategy
  5. Analytical Gradient Derivation
  6. Boundary of the Safe-Reachable Set
  7. Leibniz Integral Rule
  8. Area Gradient Calculation
  9. Pursuer's Optimal Heading
  10. Evader's Optimal Heading
  11. Conclusion
  12. Technical Appendix
  13. Proof of Lemma \ref{['lemma_Boundary_Arc_Computation']}
  14. Proof of Proposition \ref{['prop:level_set_gradients']}
  15. Proof of the Arc Centroid Formula

Key Result

Lemma 1.1

Given the area gradients $\mathbf{F}_{\mathbf{p}_i}(t)$ and $\mathbf{F}_{\mathbf{e}}(t)$, the optimal headings $\mathbf{u}_i^\star(t)$ and $\mathbf{v}^\star(t)$ that solve the min-max game eq:minmax_game are given by:

Figures (2)

  • Figure 1: Visualizing the Safe-Reachable Set $\mathcal{S}_e(t)$ for the example scenario with five pursuers and a single evader. The evader's speed is $V_e = 4$, while the pursuer speeds are given by the vector $V = [6, 6, 12, 10, 9]$. Only pursuers 2, 3 and 4 are active and contribute to the boundary.
  • Figure 2: Visualizing the Nash equilibrium for the same example scenario shown in Figure \ref{['fig:safe_set']}. Whenever a pursuer $i$ at time $t$ is inactive ($i \notin I(t)$), the pursuer is stationary; otherwise all agents follow their optimal headings according to Lemma \ref{['Nash_equilibrium']}.

Theorems & Definitions (9)

  • Lemma 1.1
  • proof
  • Lemma 2.1: Boundary Arc Computation
  • Proposition 2.2: Level-Set Gradients
  • Theorem 2.3: Geometric Form of the Area Gradients
  • proof
  • proof
  • proof
  • proof