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HOMOGENEOUS OBSERVER DESIGN 1841Appendix C. Proof of Lemma 2.13. The proof of this lemma is divided intothree parts.1. We first show, by contradiction, that there exists a real number c 0 satisfyingη 0 (θ) − cγ 0 (θ) < 0 ∀ θ ∈ S r0 , ∀ c ≥ c 0 .Suppose there is no such c 0 . This means there is a sequence (θ i ) i∈N in S r0which satisfiesη 0 (θ i ) − iγ 0 (θ i ) ≥ 0 ∀ i ∈ N .The sequence (θ i ) i∈N lives in a compact set. Thus we can extract a convergentsubsequence (θ il ) l∈N which converges to a point denoted θ ∞ .As the functions η 0 and γ 0 are bounded on S r0 and γ 0 takes nonnegativevalues, 8 γ 0 (θ il ) must go to 0 as i l goes to infinity. Since the functions η 0 andγ 0 are continuous, we get γ 0 (θ ∞ ) = 0 and η 0 (θ ∞ ) ≥ 0, which is impossible.Consequently, there exist c 0 and ε 0 > 0 such that(C.1) η 0 (θ) − cγ 0 (θ) ≤ −ε 0 < 0 ∀ θ ∈ S r0 , ∀ c ≥ c 0 .Moreover, since the functions η 0 and γ 0 are homogeneous in the standardsense (see Remark 2.6), we have the second inequality in (2.4).Following the same argument, we can find positive real numbers c ∞ and ε ∞such that(C.2) η ∞ (θ) − cγ ∞ (θ) < −ε ∞ ∀ θ ∈ S r∞ , ∀ c ≥ c ∞ ,and the third inequality in (2.4) holds.In the rest of the proof, letc 1 = max{c 0 ,c ∞ }, ε 1 = min{ε 0 ,ε ∞ } .2. Since η and γ are homogeneous in the 0-limit, there exists λ 0 such that, forall λ ∈ (0,λ 0 ] and all θ ∈ S r0 , we haveη(λ r0 ⋄ θ) ≤ λ d0 η 0 (θ) +λ d0 ε 14 , λd0 γ 0 (θ) − λ d0 ε 14c 1≤ γ(λ r0 ⋄ θ) ,which readily givesη(λ r0 ⋄ θ) − c 1 γ(λ r0 ⋄ θ) ≤ λ d0 η 0 (θ) +λ d0 ε 12 − c 1λ d0 γ 0 (θ) .Using (C.1), we getη(λ r0 ⋄ θ) − c 1 γ(λ r0 ⋄ θ) ≤ −λ d0 ε 12and therefore, since γ takes nonnegative values,∀ λ ∈ (0,λ 0 ] , ∀ θ ∈ S r0 ,η(λ r0 ⋄ θ) − cγ(λ r0 ⋄ θ) ≤−λ d0 ε 12∀ λ ∈ (0,λ 0 ] , ∀ θ ∈ S r0 , ∀ c ≥ c 1 .8 Indeed, if we had γ 0 (x) < 0 for some x in R n \{0}, by letting ɛ = − γ 0(x), the homogeneity in2the 0-limit of γ would give a real number λ> 0 satisfying γ(λr 0 ⋄x)λ d ≤ γ0 0 (x) +ɛ = γ 0(x)< 0. This2contradicts the fact that γ takes nonnegative values only. Also by continuity we have γ 0 (0) ≥ 0.Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.