Manuscript Psy03-B@1 · Psy03-B-v1.tex · lemma · 8ca3caf9

Lemma: Effective-gain continuity

Biased-noise and full-state perturbation certificates › Protection regularity and complete residuals · lines 1063–1087 · proof included or linked

Psy03-B@1#lem:gainregularity

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For $0 \leq g, g^{'} \leq G_{m a x}$ and $p, p^{'} \in [0, 1]$ ,

\abs G_{eff} (g, p) - G_{eff} (g^{'}, p^{'}) \leq \abs g - g^{'} + (G_{m a x} - c)_{+} \abs p - p^{'} .

For $a, a^{'} \in [0, a_{m a x}]$ ,

\abs q - q^{'} \leq G_{m a x} \abs a - a^{'} + a_{m a x} [\abs g - g^{'} + (G_{m a x} - c)_{+} \abs p - p^{'}] .

Proof. At fixed $p$ , the slopes in $g$ are $1$ and $1 - p$ . At fixed $g^{'}$ , the protection change has magnitude $(g^{'} - c)_{+} \abs p - p^{'}$ . Split the difference and then the product defining $q$ .

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unlabeled · line 1066
$\abs G_{eff} (g, p) - G_{eff} (g^{'}, p^{'}) \leq \abs g - g^{'} + (G_{m a x} - c)_{+} \abs p - p^{'} .$
unlabeled · line 1072
$\abs q - q^{'} \leq G_{m a x} \abs a - a^{'} + a_{m a x} [\abs g - g^{'} + (G_{m a x} - c)_{+} \abs p - p^{'}] .$

Lemma: Effective-gain continuity

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