Optimal Evasion from a Sensing-Limited Pursuer

TL;DR

This paper addresses pursuit-evasion with a sensing-limited Pursuer, formulating the Evader’s problem under a fixed final time and deriving optimal strategies in constrained and unconstrained regimes. It develops a geometric and optimal-control framework to (i) identify the no-escape region, (ii) solve for unconstrained optimal headings, and (iii) characterize the three-phase constrained trajectory that rides the Proximity Circle using an elliptic integral for the arc length. The work also analyzes a Nash/Stackelberg variant where the Pursuer selects the final time and derives a clear equilibrium pair, revealing that the Evader’s equilibrium heading is constant and independent of the initial position in that setting. Together, these results establish a rigorous baseline for intermittent sensing pursuit-evasion and guide extensions to more complex sensing-and-control strategies in partially observable environments.

Abstract

This paper investigates a partial-information pursuit evasion game in which the Pursuer has a limited-range sensor to detect the Evader. Given a fixed final time, we derive the optimal evasion strategy for the Evader to maximize its distance from the pursuer at the end. Our analysis reveals that in certain parametric regimes, the optimal Evasion strategy involves a 'risky' maneuver, where the Evader's trajectory comes extremely close to the pursuer's sensing boundary before moving behind the Pursuer. Additionally, we explore a special case in which the Pursuer can choose the final time. In this scenario, we determine a (Nash) equilibrium pair for both the final time and the evasion strategy.

Figures (7)

• Figure 1: Constrained max distance schematic. The quantities $ϕ$ and $v$ represent the Evader's effective heading and velocity in the Pursuer-fixed frame. The grey vectors demonstrate that there is another heading, $ψ$, which results in the same $ϕ$ but with a reduced effective velocity. The dashed gray circle is the locus of the end points of the resultant velocity vector $v$.
• Figure 2: Level curves of $T_{\rm{survive}}$ for $\mathbf{x}_0 \in \Omega_{\text{capture}}$.
• Figure 3: Geometric interpretation of the optimal unconstrained heading.
• Figure 4: Geometric relationship of $ϕ_{\tan}$ and the Proximity Circle entry point defined by $θ_{\tan}$.
• Figure 5: The initial positions of $E$ are divided into three regions depending on whether $E$ will be able escape $P$ and whether the escape trajectory will activate the proximity constraint. O: no-escape zone $\Omega_{\rm{capture}}$, O: Constraint is activated, O: optimal trajectories are unconstrained. For this example $T=2$ and $\mu = 0.7$.

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• Lemma 3
• Remark 1
• Lemma 4
• Lemma 5
• Remark 2
• Theorem 1
• Theorem 2
• Proposition 1