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LIT-1893: Input-to-State Safety With Control Barrier Functions

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Properties5
StatusActive — read in full 2026-10-31 (NOTE-1477); worth reading as the short statement of graded robust safety: under a bounded input disturbance a barrier-function controller keeps the state not in the safe set but in a set enlarged by a class-K function of the disturbance, and a damping term along the barrier's input direction delivers this with an enlargement of ‖d‖²/(4ε) (THEORY-1267). It is not the robust box invariance CASE-190 certifies, which is THEORY-134's.
Version2
Tagscontrol-theory, mathematics, complex-systems
Cited as “paper” byNOTE-1477 — Kolathaya & Ames — input-to-state safe control barrier functions
Cited as “source” byTHEORY-1267 — A safe set that a nominal barrier-function controller keeps invariant can be left under an arbitrarily small input disturbance, but if the barrier inequality holds with a disturbance penalty, ḣ ≥ −α(h) − ι(‖d‖), the state stays in a larger set {h + γ(‖d‖∞) ≥ 0} with γ class-K, and adding a damping term along the barrier's input direction to any barrier controller gives this with an enlargement of ‖d‖²/(4ε) in units of h

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Kolathaya, S., & Ames, A. D. (2019). Input-to-state safety with control barrier functions. IEEE Control Systems Letters, 3(1), 108–113.

Read in full in NOTE-1477.

Why this is here

The one control-theory work the exchange gave an identifier for, when the owner said at U17 “I bet control theory has a bunch of tools that would be useful to us”. A17 used barrier certificates for the viable boxes of its constructions (CASE-190). The record’s nearest tools are Aubin’s viability theory (THEORY-134) and contraction under singular perturbation (THEORY-771). The reading separates two things the case runs together. The case’s certificate shows each box face’s inward margin exceeds the disturbance bound. That is robust invariance of the box itself, which is viability. ISSf is the graded guarantee for disturbances beyond the margin: the state leaves the box, but only by γ(‖d‖) (THEORY-1267). The topic vocabulary has no word for control theory; that is left to the owner.

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