| Abstract (2007) | |||||||||||||||
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| Lemma 5 in [2] was incorrect. A correction is provided here. The error does not affect the rest of the results in the paper. 1 The Error and the Fix There is an error in the statement and the proof of Lemma 5 in the paper [2]. The estimate about α0(|z(t)|) towards the end of the proof of Lemma 5 is incorrect. It should be replaced by � τ(t) α0(|z(t)|) ≤ β(|z0 | , τ(t)) + γ(|dx(s)|) ds 0 � t ≤ β(|z0 | , τ(t)) + γ(|d(s)|)ϕ(z(s)) ds. Without knowing whether ϕ(|z(s)|) is bounded, one cannot conclude that α0(|z(t)|) is bounded. Below we provide a correction of the lemma by posing a boundedness condition on the function ϕ. Lemma 1 Consider a general system 0 ˙x = f(x, d), (1) where f is a locally Lipschitz function and d is a locally essentially bounded disturbance input. Suppose the system is iISS. Let ϕ: R n → R be a bounded, positive definite and smooth function. Then the system is iISS. ˙x = ϕ(x)f(x, d) (2) Remark 1 The change in the lemma does not affect the consequent results in [2]. This is because in Lemma 6 we indeed require that the function ϕ be bounded by 1. Hence, the main result in Section 8 of [2], Theorem 1, remains valid. | |||||||||||||||
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