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Guarding a Target Area from a Heterogeneous Group of Cooperative Attackers

Yoonjae Lee, Goutam Das, Daigo Shishika, Efstathios Bakolas

TL;DR

This work addresses a dynamic guard–attack game with one defender and an arbitrary, heterogeneous attacker team aiming to reach a convex target area. It develops a geometric, phase-based optimization framework that reduces to a single-level problem over time-extended capture points, and proves equilibrium policies via Isaacs equation analysis and Fiacco sensitivity results. The approach handles heterogeneity in attacker speeds and resource weights, and identifies conditions under which the value function is differentiable and the equilibrium controls are unique, with nonconvex feasible regions managed through convex-concave techniques. Numerical experiments with six attackers illustrate emergent cooperative behaviors driven by heterogeneity, such as sacrificial actions by lower-weight attackers to aid others. Overall, the paper generalizes prior homogeneous/limited-attacker results to arbitrary numbers and target shapes, providing a rigorous method and validation for heterogeneous target-guarding games with practical implications for autonomous defense systems.

Abstract

In this paper, we investigate a multi-agent target guarding problem in which a single defender seeks to capture multiple attackers aiming to reach a high-value target area. In contrast to previous studies, the attackers herein are assumed to be heterogeneous in the sense that they have not only different speeds but also different weights representing their respective degrees of importance (e.g., the amount of allocated resources). The objective of the attacker team is to jointly minimize the weighted sum of their final levels of proximity to the target area, whereas the defender aims to maximize the same value. Using geometric arguments, we construct candidate equilibrium control policies that require the solution of a (possibly nonconvex) optimization problem. Subsequently, we validate the optimality of the candidate control policies using parametric optimization techniques. Lastly, we provide numerical examples to illustrate how cooperative behaviors emerge within the attacker team due to their heterogeneity.

Guarding a Target Area from a Heterogeneous Group of Cooperative Attackers

TL;DR

This work addresses a dynamic guard–attack game with one defender and an arbitrary, heterogeneous attacker team aiming to reach a convex target area. It develops a geometric, phase-based optimization framework that reduces to a single-level problem over time-extended capture points, and proves equilibrium policies via Isaacs equation analysis and Fiacco sensitivity results. The approach handles heterogeneity in attacker speeds and resource weights, and identifies conditions under which the value function is differentiable and the equilibrium controls are unique, with nonconvex feasible regions managed through convex-concave techniques. Numerical experiments with six attackers illustrate emergent cooperative behaviors driven by heterogeneity, such as sacrificial actions by lower-weight attackers to aid others. Overall, the paper generalizes prior homogeneous/limited-attacker results to arbitrary numbers and target shapes, providing a rigorous method and validation for heterogeneous target-guarding games with practical implications for autonomous defense systems.

Abstract

In this paper, we investigate a multi-agent target guarding problem in which a single defender seeks to capture multiple attackers aiming to reach a high-value target area. In contrast to previous studies, the attackers herein are assumed to be heterogeneous in the sense that they have not only different speeds but also different weights representing their respective degrees of importance (e.g., the amount of allocated resources). The objective of the attacker team is to jointly minimize the weighted sum of their final levels of proximity to the target area, whereas the defender aims to maximize the same value. Using geometric arguments, we construct candidate equilibrium control policies that require the solution of a (possibly nonconvex) optimization problem. Subsequently, we validate the optimality of the candidate control policies using parametric optimization techniques. Lastly, we provide numerical examples to illustrate how cooperative behaviors emerge within the attacker team due to their heterogeneity.
Paper Structure (10 sections, 2 theorems, 32 equations, 4 figures)

This paper contains 10 sections, 2 theorems, 32 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notation
  4. Problem Statement
  5. Preliminary: Single-Attacker Game
  6. Main Results
  7. Optimization Formulation of a Geometric Solution
  8. Proof of Equilibrium
  9. Numerical Example
  10. Conclusion

Key Result

Theorem 1

Let $\bar{\mathbf{x}} \in \mathcal{X}$. Let $\mathbf{p}^*$ be a local minimizer of problem eq:main_prob at $\bar{\mathbf{x}}$. If there exists a multiplier $\lambda^*$ such that the triple $(\mathbf{p}^*,\lambda^*,\bar{\mathbf{x}})$ satisfies all of the conditions below: then there exists a neighborhood $\mathcal{B}(\bar{\mathbf{x}},\epsilon) = \{\mathbf{x} : \|\mathbf{x}-\bar{\mathbf{x}}\|<\epsi

Figures (4)

  • Figure 1: Game of protecting a target area ($\mathcal{T}$) from attackers ($A_i$) with heterogeneous speed ratios ($\nu_i$) and weights ($\theta_i$). The dashed lines depict the order of capture.
  • Figure 2: Selected parameter values.
  • Figure 3: Equilibrium trajectory of the game. The blue and the red markers indicate the defender and the attackers, respectively. The pink regions illustrate the (time-extended) safe-reachable regions of the attackers, whereas the grey circle illustrates the boundary of the target area.
  • Figure 4: Proximity levels of optimal capture points for $10^3$ randomly generated initial attacker positions.

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1: fiacco
  • Theorem 2
  • proof
  • Remark 5
  • Remark 6