Characterization of Optimal Trajectories in ℝ3

Abstract

We characterize the set of all trajectories $$\mathcal{T}$$ of an object t moving in a given corridor Y that are furthest away from a family $$\mathbb{ S} =\{ S\} $$ of fixed unfriendly observers. Each observer is equipped with a convex open scanning cone K(S) with vertex S. There are constraints on the multiplicity of covering the corridor Y by the cones K and on the “thickness” of the cones. In addition, pairs S, S′ for which [S, S′] ⊂ (K(S) ∩ K(S′)) are not allowed. The original problem $$\max\nolimits _{\mathcal{T}} \min \{d(t, S): t \in \mathcal{T}, S \in \mathbb{S}\}$$ , where d(t, S) = ∥t − S∥ for t ∈ K(S) and d(t,S) = +∞ for t ∉ K(S), is reduced to the problem of finding an optimal route in a directed graph whose vertices are closed disjoint subsets (boxes) from $$Y\backslash { \cup _S}K\left( S \right)$$ . Neighboring (adjacent) boxes are separated by some cone K(S). An edge is a part $${\cal T}\left( S \right)$$ of a trajectory $${\cal T}$$ that connects neighboring boxes and optimally intersects the cone K(S). The weight of an edge is the deviation of S from $${\cal T}\left( S \right)$$ . A route is optimal if it maximizes the minimum weight.