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1830 V. ANDRIEU, L. PRALY, AND A. ASTOLFIis homogeneous in the bi-limit with weights (r 0,1 ,...,r 0,i ), (r ∞,1 ,...,r ∞,i ), degreesd V0 − r 0,j , d V∞ − r ∞,j , and approximating functions ∂Vi,0. Moreover, we havefor all X i ∈ R i \{0},∂X j, ∂Vi,∞∂X j(4.7)∂V i∂X i(X i )[S i X i + B i φ i (X i )] < 0 ,∂V i,0(X i )[S i X i + B i φ i,0 (X i )] < 0 ,∂X i∂V i,∞(X i )[S i X i + B i φ i,∞ (X i )] < 0 .∂X iFollowing [21], consider the Lyapunov function V i+1 : R i+1 → R + defined by∫ X i+1(d V0 −r 0,i+1d V0 −r 0,i+1)rV i+1 (X i+1 ) = V i (X i )+ h 0,i+1 r− φ i (X i ) 0,i+1dhφ i(X i)∫ X i+1(d V∞ −r ∞,i+1d V∞ −r ∞,i+1)r+ h ∞,i+1 r− φ i (X i ) ∞,i+1dh .φ i(X i)This function is positive definite and proper. Furthermore, as d V∞ and d V0 satisfy(4.6), we haved V∞ − r ∞,i+1r ∞,i+1≥ d V 0− r 0,i+1r 0,i+1>α i ≥ 1 .Since the function ψ i (X i )=φ i (X i ) αi is C 1 , this inequality yields that the functionV i+1 is C 1 . Finally, for each j in {1,...,n}, the function ∂Vi+1is homogeneous in∂X jthe bi-limit with associated triples((r 0,1 ,...,r 0,i+1 ),d V0 − r 0,j , ∂V ) (i+1,0, (r ∞,1 ,...,r ∞,i+1 ),d V∞ − r ∞,j , ∂V )i+1,∞.ψ i+1∂X jStep 2. Definition of the control law. Recall (1.6) and consider the function: R i+1 → R defined by∫ αXii+1 −φi(Xi)α i (ψ i+1 (X i+1 )=−kH |s| αi+1 d 0 +r 0,i+10α i r 0,i+1−1, |s|α i+1d∞+r ∞,i+1where k in R + is a design parameter and α i+1 is selected as{ }αi r 0,i+1 α i r ∞,i+1α i+1 ≥ max,, 1d 0 + r 0,i+1 d ∞ + r ∞,i+1α i r ∞,i+1.∂X j)−1ds ,ψ i+1 takes values with the same sign as X i+1 − φ i (X i ), is C 1 , and, by Proposition2.12, is homogeneous in the bi-limit. Furthermore, by Proposition 2.10, for each jin {1,...,i +1}, the function ∂ψi+1is homogeneous in the bi-limit, with weights∂X j(r 0,1 ,...,r 0,i+1 ), (r ∞,1 ,...,r ∞,i+1 ), degrees α i+1 (r 0,i+1 + d 0 ) − r 0,j , α i+1 (r ∞,i+1 +d ∞ ) − r ∞,j , and approximating functions ∂ψi+1,0 ∂ψ,i+1,∞. With this at hand, we∂X j ∂X jchoose the control law φ i+1 asφ i+1 (X i+1 ) = ψ i+1 (X i+1 ) 1α i+1 .Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.