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Evasive Path Planning Under Surveillance Uncertainty

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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.

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Notes

  1. Fast sweeping [39] is another popular approach for gaining efficiency in solving Eikonal equations. We refer readers to [7, 8] for a review of many other “fast” techniques, including the hybrid marching/sweeping methods that aim to combine the best features of both approaches. Even though our own implementation is based on fast marching, any of these methods can be used to solve isotropic control problems arising in subsequent sections. Which one will be faster depends on the domain geometry and the particular pointwise observability functions.

  2. Here, we describe these methods in terms of exposure to different observer’s positions, but both of them were introduced for much more general multiobjective control problems. In many applications, it is necessary to balance completely different criteria, e.g., time vs fuel vs money vs threat, etc. Other methods for approximating the full PF can be found in [13] and [18].

  3. This result assumes that the set \(S=\{ (s_1, s_2, \ldots , s_r) \mid s_i = P({\hat{\varvec{x}}}_i, \varvec{a}( \cdot )); i = 1,2,\dots ,r; \varvec{a}( \cdot )\in {\mathcal {A}}\} \subset {\mathbb {R}}^r \) is bounded and co(S) is closed. In our case, S is not bounded for the full set of control functions in \({\mathcal {A}}\) but becomes bounded if we restrict our attention to Pareto optimal control functions.

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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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Authors and Affiliations

  1. Center for Applied Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Marc Aurèle Gilles

  2. Department of Mathematics, Center for Applied Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Alexander Vladimirsky

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  1. Marc Aurèle Gilles
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  2. Alexander Vladimirsky
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Correspondence to Alexander Vladimirsky.

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Gilles, M.A., Vladimirsky, A. Evasive Path Planning Under Surveillance Uncertainty. Dyn Games Appl 10, 391–416 (2020). https://doi.org/10.1007/s13235-019-00327-x

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