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Evasive Path Planning Under Surveillance Uncertainty

  • Marc Aurèle Gilles
  • Alexander VladimirskyEmail author
Article
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Abstract

The classical setting of optimal control theory assumes full knowledge of the process dynamics and the costs associated with every control strategy. The problem becomes much harder if the controller only knows a finite set of possible running cost functions, but has no way of checking which of these running costs is actually in place. In this paper we address this challenge for a class of evasive path planning problems on a continuous domain, in which an evader needs to reach a target while minimizing his exposure to an enemy observer, who is in turn selecting from a finite set of known surveillance plans. Our key assumption is that both the evader and the observer need to commit to their (possibly probabilistic) strategies in advance and cannot immediately change their actions based on any newly discovered information about the opponent’s current position. We consider two types of evader behavior: in the first one, a completely risk-averse evader seeks a trajectory minimizing his worst-case cumulative observability, and in the second, the evader is concerned with minimizing the average-case cumulative observability. The latter version is naturally interpreted as a semi-infinite strategic game, and we provide an efficient method for approximating its Nash equilibrium. The proposed approach draws on methods from game theory, convex optimization, optimal control, and multiobjective dynamic programming. We illustrate our algorithm using numerical examples and discuss the computational complexity, including for the generalized version with multiple evaders.

Keywords

Path planning Semi-infinite games Nash equilibrium Surveillance evasion Convex optimization Hamilton–Jacobi PDEs 

Mathematics Subject Classification

49N75 49N90 49K35 91A05 90C29 

Notes

Acknowledgements

The authors would like to thank Alex Townsend and anonymous reviewers for their helpful suggestions.

Funding

This work is supported in part by the National Science Foundation Grant DMS-1738010. The second author’s work is also supported by the Simons Foundation Fellowship.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Applied MathematicsCornell UniversityIthacaUSA
  2. 2.Department of Mathematics, Center for Applied MathematicsCornell UniversityIthacaUSA

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