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HOMOGENEOUS OBSERVER DESIGN 1819then the function Φ i (x) = ∫ x iφ(x0 1 ,...,x i−1 , s, x i+1 ,...,x n ) ds is homogeneous inthe 0- (resp., ∞-) limit with associated triple (r 0 ,d 0 + r 0,i , Φ i,0 ) (resp., (r ∞ ,d ∞ +r ∞,i , Φ i,∞ )), with Φ i,0 (x) = ∫ x iφ0 0 (x 1 ,...,x i−1 , s, x i+1 ,...,x n ) ds (resp., Φ i,∞ (x) =∫ xiφ0 ∞ (x 1 ,...,x i−1 , s, x i+1 ,...,x n ) ds).By exploiting the definition of homogeneity in the bi-limit, it is possible to establishresults which are straightforward extensions of well-known results based on thestandard notion of homogeneity. These results are given as corollaries of the followingkey technical lemma, the proof of which is given in Appendix C.Lemma 2.13 (key technical lemma). Let η : R n → R and γ : R n → R + be twofunctions homogeneous in the bi-limit, with weights r 0 and r ∞ , degrees d 0 and d ∞ ,and approximating functions, η 0 , η ∞ and γ 0 , γ ∞ such that the following hold:{ x ∈ R n \{0} : γ(x) =0} ⊆{ x ∈ R n : η(x) < 0 } ,{ x ∈ R n \{0} : γ 0 (x) =0} ⊆{ x ∈ R n : η 0 (x) < 0 } ,{ x ∈ R n \{0} : γ ∞ (x) =0} ⊆{ x ∈ R n : η ∞ (x) < 0 } .Then there exists a real number c ∗ such that, for all c ≥ c ∗ and for all x in R n \{0},(2.4)η(x) − cγ(x) < 0 , η 0 (x) − cγ 0 (x) < 0 , η ∞ (x) − cγ ∞ (x) < 0 .Example 2.14. To illustrate the importance of this lemma, consider, for (x 1 ,x 2 )in R 2 , the functionsη(x 1 ,x 2 )=x 1 x 2 −|x 1 | r 1 +r 2r 1 , γ(x 1 ,x 2 )=|x 2 | r 1 +r 2r 2 ,with r 1 > 0 and r 2 > 0. They are homogeneous in the standard sense, and thereforein the bi-limit, with the same weight r =(r 1 ,r 2 ) and the same degree d = r 1 + r 2 .Furthermore, the function γ takes positive values, and for all (x 1 ,x 2 ) in {(x 1 ,x 2 ) ∈R 2 \{0} : γ(x 1 ,x 2 )=0} we haveη(x 1 ,x 2 ) = −|x 1 | r 1 +r 2r 1 < 0 .Thus Lemma 2.13 yields the existence of a positive real number c ∗ such that for allc ≥ c∗, we have(2.5) x 1 x 2 −|x 1 | r 1 +r 2r 1 − c |x 2 | r 1 +r 2r 2 < 0 ∀ (x 1 ,x 2 ) ∈ R 2 \{0} .This is a generalization of the procedure known as the completion of the squares inwhich, however, the constant c ∗ 1 is not specified.Corollary 2.15. Let φ : R n → R and ζ : R n → R + be two homogeneous in thebi-limit functions with the same weights r 0 and r ∞ , degrees d φ,0 , d φ,∞ and d ζ,0 , d ζ,∞ ,and approximating functions η 0 , φ ∞ and ζ 0 , ζ ∞ . If the degrees satisfy d φ,0 ≥ d ζ,0 andd φ,∞ ≤ d ζ,∞ , and the functions ζ, ζ 0 and ζ ∞ are positive definite, then there exists apositive real number c satisfyingProof. Consider the two functionsφ(x) ≤ cζ(x) ∀ x ∈ R n .η(x) := φ(x)+ζ(x), γ(x) := ζ(x) .Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.