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Safe Control for Pursuit-Evasion with Density Functions

Mustafa Bozdag, Arya Honarpisheh, Mario Sznaier

TL;DR

This work addresses safe control in pursuit–evading scenarios with dynamic unsafe regions by formulating a density-function based robust safe-control problem. By treating the pursuer as a bounded disturbance and leveraging Liouville-based density certificates, the authors avoid solving the Hamilton–Jacobi–Isaacs PDE and instead solve a convex SOS program to synthesize both the density function and the evader control. The approach yields sufficient conditions for weak eventuality and evasion expressed through polynomial inequalities over semi-algebraic sets, enabling scalable computation. Numerical experiments with two pursuer strategies demonstrate that the evader can reach a designated target while remaining outside the dynamic capture region, underscoring the practical viability of the method.

Abstract

This letter presents a density function based safe control synthesis framework for the pursuit-evasion problem. We extend safety analysis to dynamic unsafe sets by formulating a reach-avoid type pursuit-evasion differential game as a robust safe control problem. Using density functions and semi-algebraic set definitions, we derive sufficient conditions for weak eventuality and evasion, reformulating the problem into a convex sum-of-squares program solvable via standard semidefinite programming solvers. This approach avoids the computational complexity of solving the Hamilton-Jacobi-Isaacs partial differential equation, offering a scalable and efficient framework. Numerical simulations demonstrate the efficacy of the proposed method.

Safe Control for Pursuit-Evasion with Density Functions

TL;DR

This work addresses safe control in pursuit–evading scenarios with dynamic unsafe regions by formulating a density-function based robust safe-control problem. By treating the pursuer as a bounded disturbance and leveraging Liouville-based density certificates, the authors avoid solving the Hamilton–Jacobi–Isaacs PDE and instead solve a convex SOS program to synthesize both the density function and the evader control. The approach yields sufficient conditions for weak eventuality and evasion expressed through polynomial inequalities over semi-algebraic sets, enabling scalable computation. Numerical experiments with two pursuer strategies demonstrate that the evader can reach a designated target while remaining outside the dynamic capture region, underscoring the practical viability of the method.

Abstract

This letter presents a density function based safe control synthesis framework for the pursuit-evasion problem. We extend safety analysis to dynamic unsafe sets by formulating a reach-avoid type pursuit-evasion differential game as a robust safe control problem. Using density functions and semi-algebraic set definitions, we derive sufficient conditions for weak eventuality and evasion, reformulating the problem into a convex sum-of-squares program solvable via standard semidefinite programming solvers. This approach avoids the computational complexity of solving the Hamilton-Jacobi-Isaacs partial differential equation, offering a scalable and efficient framework. Numerical simulations demonstrate the efficacy of the proposed method.

Paper Structure

This paper contains 15 sections, 3 theorems, 32 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Notation
  4. Safety and Eventuality
  5. Pursuit-Evasion Games
  6. Density Functions
  7. Sum-of-Squares
  8. Problem Statement
  9. Density-Based Robust Evasion
  10. Convex Relaxation
  11. System Dynamics and Semi-Algebraic Sets
  12. Designing the Controller
  13. SOS Implementation
  14. NUMERICAL EXAMPLE
  15. CONCLUSION

Key Result

Theorem 1

From prajna2007convex. Consider the dynamical system $\mathbf{\dot{x}}=f(\mathbf{x})$ with $f\in \mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$, the bounded set $\mathcal{X}\subset\mathbb{R}^n$, and the sets $\mathcal{X}_i,\mathcal{X}_a,\mathcal{X}_r\subseteq \mathcal{X}$. If there exists an open set $\T then the weak eventuality and safety properties hold. For almost all initial states $\mathbf{x}_0\i

Figures (3)

  • Figure 1: The reach-avoid environment. The objective is to remain in $\mathcal{X}$, avoiding $\mathcal{X}_a$, and reaching $\mathcal{X}_r$ using a connected path within $\mathcal{X}_c$. The initial states for the evader and the pursuer are indicated with $\mathcal{X}_{ie}$ and $\mathcal{X}_{ip}$, which together form $\mathcal{X}_{i}=\mathcal{X}_{ie}\cap\mathcal{X}_{ip}$. For pursuit-evasion, the unsafe set $\mathcal{X}_a$ is dynamic. It is important to note that all these sets are defined in a four-dimensional state space. The 2D illustration provided is for intuition only and does not fully capture the true structure of these sets.
  • Figure 2: Final trajectories with tail-chasing (left) and go-to-middle (right) pursuers. The evader reaching the target set while avoiding the catch radius of the pursuer and remaining in the bounded region, using the density function $\rho$. The dashed black line is the $\rho=0$ level set.
  • Figure 3: The distance between the evader and the pursuer over time for tail-chasing (left) and go-to-middle (right) pursuers. The dashed red line shows the catch-radius of the pursuer.

Theorems & Definitions (8)

  • Theorem 1
  • Remark 1
  • Definition 1
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • Theorem 3