By Pierre Lelong

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Then there exists a continuous function φ2 (·) in [0, a) such that φ1 (s) = φ2 (s)s, s ∈ [0, a) Proof (i) Suppose that β(x, s) : Rn × R + → R+ is a class K I function. 2 Comparison Functions 31 (ii) Since the function φ1 (·) is a C 1 function in [0, a), its derivative continuous in [0, a). 4) From the definition of φ2 (·), it is clear to see that (1) (2) if s = 0, then φ1 (s) = φ2 (s)s; if s = 0, then from φ1 (0) = 0, φ1 (s) = φ2 (s)s. Therefore, the expression φ1 (s) = φ2 (s)s holds for s ∈ [0, a).

A Lipschitz function may not be differentiable and a simple example is the scalar function f (x) = |x| at the origin x = 0 in x ∈ R. 2) is not Lipschitz in the compact set x ∈ [0, 1] for any constant α satisfying 1 < α < 2. 2) is not bounded in the interval [0, 1]. 1 ([91]) Consider a function f (x) : Rn → Rm which is differentiable in the domain Ω. 1) holds. 2 Generalised Lipschitz Condition The well-known Lipschitz condition in Sect. 1 will be extended to a more general case which will be used later in the analysis.

If the uncertainty or disturbance acts in the input/control channel or the effects are equivalent to an uncertainty acting in the input channel, it is called matched uncertainty. Otherwise it is called mismatched uncertainty. 7) experiences uncertainties φ(t, x) and ψ(t, x) described by x˙ = F(t, x) + G(t, x)(u + φ(t, x)) + ψ(t, x). 23) Then, the term φ(t, x) is called matched uncertainty. In addition, if the uncertainty ψ(t, x) can be modelled as ψ(t, x) = G(t, x)χ (t, x) where χ (·) represents the uncertainty, it is clear to see that the uncertainty of the term ψ(·) is reflected by the uncertainty χ (·) which is exactly acting in the input channel.