THÈSE DE DOCTORAT Ecole Doctorale « Sciences et ...

Annexe D. Théorèmes du petit gain pour des systèmes paramétrés 195
Par conséquent, d’après (D.80), (D.97), (D.98), (D.99), (D.100) et (D.101) :
{
|ω 1 (x 1 (t))| ≤ max β 1 (|ω 1 (x 1 ( t 2 ))|, t 2 ),γωa 2
1 (‖ωa 2
2(x 2 )‖ [
t
2 ,t)),γωb 1 (‖ωb 2(x 2 )‖ [
t
,t)), 2
≤
γ1 u (‖u 1 ‖ [
t
{
max β 1 (β 1 (|ω 1 (x 1 (t 0 ))|,0), t 2 ),β 1(γ ωa 2
1 (β 2(|ω 2 (x 2 (t 0 ))|,0)), t 2 ),
2 ,t)) }
(D.102)
β 1 (γ ωa 2
1 (γu 2 (‖u 2‖ [t0 ,t) )), t 2 ),β 1(γ u 1 (‖u 1‖ [t0 ,t) ), t 2 ),
β 1 (γ ωb 2
1 (α 2(|ω b 2 (x 2(t 0 ))|)), t 2 ),
β 1 (γ ωb 2
1 (ηωa 2
2 (β 2(|ω 2 (x 2 (t 0 ))|,0))), t 2 ),
β 1 (γ ωb 2
1 (ηωa 2
2 (γu 2 (‖u 2‖ [t0 ,t) ))), t 2 ),
β 1 (γ ωb 2
1 (ηu 2 (‖u 2‖ [t0 ,t) )), t 2 ),β 1(m, t 2 ),
γ ωa 2
1 (β 2(2β 2 (|ω 2 (|x 2 (t 0 ))|,0), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (β 1(|ω 1 (x 1 (t 0 ))|,0)), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γu 1 (‖u 1‖ [t0 ,t) )), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (α 2(|ω b 2 (x 2(t 0 ))|))), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (ηω 1
2 (β 1(|ω 1 (x 1 (t 0 ))|,0)))), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (ηω 1
2 (γωa 2
1 (β 2(|ω 2 (x 2 (t 0 ))|,0))))), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (ηω 1
2 (γωa 2
1 (γu 2 (‖u 2‖ [t0 ,t) ))))), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (ηω 1
2 (γu 1 (‖u 1‖ [t0 ,t) )))), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (ηω 1
2 (γωb 2
1 (α 2(|ω b 2 (x 2(t 0 ))|))))), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (ηω 1
2 (γωb 2
1 (ηωa 2
2 (β 2(|ω 2 (x 2 (t 0 ))|,0)))))), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (ηω 1
2 (γωb 2
1 (ηωa 2
2 (γu 2 (‖u 2‖ [t0 ,t) )))))), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (ηω 1
2 (γωb 2
1 (ηu 2 (‖u 2‖ [t0 ,t) ))))), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (ηω 1
2 (m))), t 2 )),
γ ωa 2
1 (β 2(2γ ω 1
2 (γωb 2
1 (ηu 2 (‖u 2‖ [t0 ,t) ))), t 2 )),γωa 2
1 (β 2(2γ u 2 (‖u 2‖ [t0 ,t) ), t 2 )),
γ ωa 2
1 (β 2(2m, t 2 )),γωa 2
1 (β 2(2α 2 (|ω b 2 (x 2(t 0 ))|), t 2 )),
γ ωa 2
1 (β 2(2η ω 1
2 (β 1(|ω 1 (x 1 (t 0 ))|,0)), t 2 )),