Sounds like a distribution except that instead of having linear subspaces you have cones. There's this paper: Langerock, "Conic Distributions and Accessible Sets,"Langerock, "Conic Distributions and Accessible Sets," but it sounds an awful lot like your question (and I wonder if that's where you're starting from in the first place!). It also doesn't say anything about the computability of the accessible set, though they do provide some characterization.
Glorfindel
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