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Bang-Bang Evasion: Its Stochastic Optimality and a Terminal-Set-Based Implementation

Liraz Mudrik, Yaakov Oshman

TL;DR

The paper studies optimal evasion in planar endgames with a bounded-acceleration evader facing a linear-guidance interceptor under stochastic information. It proves that an optimal evasion policy exists and that at least one optimal policy is bang-bang, extending deterministic results to uncertainty via the generalized separation theorem. It then proposes a terminal-set-based evasion (TSE) strategy that uses posterior state estimates to compare terminal outcomes and select between extreme accelerations, reducing the problem to a finite set of decisions. Through Monte Carlo experiments against a PN pursuer and comparisons with RTS, Singer, and weaving models, TSE achieves larger miss distances and better evasion performance, offering a practically scalable solution for stochastic endgames.

Abstract

We address the problem of optimal evasion in a planar endgame engagement, where a target with bounded lateral acceleration seeks to avoid interception by a missile guided by a linear feedback law. Contrary to existing approaches, that assume perfect information or use heuristic maneuver models in stochastic settings, we formulate the problem in an inherently stochastic framework involving imperfect information and bounded controls. Complying with the generalized separation theorem, the control law factors in the posterior distribution of the state. Extending the well-known optimality of bang-bang evasion maneuvers in deterministic settings to the realm of realistic, stochastic evasion scenarios, we firstly prove that an optimal evasion strategy always exists, and that the set of optimal solutions includes at least one bang-bang policy, rendering the resulting optimal control problem finite-dimensional. Leveraging this structure, we secondly propose the closed-loop terminal-set-based evasion (TSE) strategy, and demonstrate its effectiveness in simulation against a proportional navigation pursuer. Monte Carlo simulations show that the TSE strategy outperforms traditional stochastic evasion strategies based on random telegraph, Singer, and weaving models.

Bang-Bang Evasion: Its Stochastic Optimality and a Terminal-Set-Based Implementation

TL;DR

The paper studies optimal evasion in planar endgames with a bounded-acceleration evader facing a linear-guidance interceptor under stochastic information. It proves that an optimal evasion policy exists and that at least one optimal policy is bang-bang, extending deterministic results to uncertainty via the generalized separation theorem. It then proposes a terminal-set-based evasion (TSE) strategy that uses posterior state estimates to compare terminal outcomes and select between extreme accelerations, reducing the problem to a finite set of decisions. Through Monte Carlo experiments against a PN pursuer and comparisons with RTS, Singer, and weaving models, TSE achieves larger miss distances and better evasion performance, offering a practically scalable solution for stochastic endgames.

Abstract

We address the problem of optimal evasion in a planar endgame engagement, where a target with bounded lateral acceleration seeks to avoid interception by a missile guided by a linear feedback law. Contrary to existing approaches, that assume perfect information or use heuristic maneuver models in stochastic settings, we formulate the problem in an inherently stochastic framework involving imperfect information and bounded controls. Complying with the generalized separation theorem, the control law factors in the posterior distribution of the state. Extending the well-known optimality of bang-bang evasion maneuvers in deterministic settings to the realm of realistic, stochastic evasion scenarios, we firstly prove that an optimal evasion strategy always exists, and that the set of optimal solutions includes at least one bang-bang policy, rendering the resulting optimal control problem finite-dimensional. Leveraging this structure, we secondly propose the closed-loop terminal-set-based evasion (TSE) strategy, and demonstrate its effectiveness in simulation against a proportional navigation pursuer. Monte Carlo simulations show that the TSE strategy outperforms traditional stochastic evasion strategies based on random telegraph, Singer, and weaving models.

Paper Structure

This paper contains 20 sections, 4 theorems, 48 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Linearized Kinematics and Dynamics
  4. Engagement Duration
  5. Estimation
  6. Pursuer Guidance Strategy
  7. Linear Guidance Laws
  8. Information About The Interceptor's States
  9. Performance Index
  10. Optimal Evasion Structure
  11. Terminal-Set-Based Evasion
  12. Scope and Assumptions
  13. Terminal-Set Representation
  14. Evasion Law Derivation
  15. Discussion
  16. ...and 5 more sections

Key Result

Theorem III.1

At each time $t^n$, there exists at least one optimal control sequence for the evasion problem. Furthermore, of all optimal control sequences, at least one must possess the well-known bang-bang structure, that is for all $k=n,\dots,k'-1$.

Figures (4)

  • Figure 1: Planar engagement geometry
  • Figure 2: Expected cost $J(u_{T}^{n})$ under uniform future input model. The optimal control lies at the control bounds, confirming the bang-bang structure.
  • Figure 3: Evader acceleration command profiles for a representative engagement under the TSE (blue line), RTS (red line), Singer (magenta line), and weaving (black line) evasion strategies.
  • Figure 4: Empirical CDF of miss distance across 10,000 MC trials using the TSE (blue line), RTS (red line), Singer (magenta line), and weaving (black line) evasion strategies.

Theorems & Definitions (9)

  • Example II.1
  • Example II.2
  • Theorem III.1
  • proof
  • Theorem III.2: Theorem 7.42 in beck_introduction_2014
  • Lemma III.3
  • proof
  • Lemma III.4
  • proof