THÈSE DE DOCTORAT Ecole Doctorale « Sciences et ...

Annexe D. Théorèmes du petit gain pour des systèmes paramétrés 199
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γu 1 (‖u 1‖ [t0 ,t) ))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γωb 2
1 (α 2(|ω b 2(x 2 (t 0 ))|)))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γωb 2
1 (ηω 1
2 (β 1(|ω 1 (x 1 (t 0 ))|,0))))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γωb 2
1 (ηω 1
2 (γωa 2
1 (β 2(|ω 2 (x 2 (t 0 ))|,0)))))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γωb 2
1 (ηω 1
2 (γωa 2
1 (γu 2 (‖u 2 ‖ [t0 ,t) )))))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γωb 2
1 (ηω 1
2 (γu 1 (‖u 1 ‖ [t0 ,t) ))))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γωb 2
1 (ηω 1
2 (γωb 2
1 (α 2(|ω b 2(x 2 (t 0 ))|)))))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 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 ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γωb 2
1 (ηω 1
2 (γωb 2
1 (ηωa 2
2 (γu 2 (‖u 2 ‖ [t0 ,t) ))))))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γωb 2
1 (ηω 1
2 (γωb 2
1 (ηu 2 (‖u 2 ‖ [t0 ,t) )))))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γωb 2
1 (ηω 1
2 (m)))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γω 1
2 (γωb 2
1 (ηu 2 (‖u 2 ‖ [t0 ,t) )))), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (γu 2 (‖u 2 ‖ [t0 ,t) )), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η ωa 2
2 (m), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (β 2(2η u 2 (‖u 2 ‖ [t0 ,t) ), t 2 ))),
γ ωb 2
1 (ηωa 2
2 (γω 1
2 (‖ω 1(x 1 )‖ [
t
2 ,t)))),
γ ωb 2
1 (ηωa 2
2 (γu 2 (‖u 2‖ [
t
2 ,t)))),
γ ωb 2
1 (ηu 2 (‖u 2 ‖ [t0 ,t) )),γu 1 (‖u 1 ‖ [
t
2 ,t)) }
, (D.103)
que l’on écrit :
|ω 1 (x 1 (t))| ≤ max
{
˜β1 (max { |ω 1 (x 1 (t 0 ))|,|ω 2 (x 2 (t 0 ))| } ,t),
˜σ u 1 (max { ‖u 1 ‖ [t0 ,t) , ‖u 2‖ [t0 ,t)
}
),γ
ω a 2
1 (γω 1
2 (‖ω 1(x 1 )‖ [
t
2 ,t))),
γ ωb 2
1 (ηω 1
2 (‖ω 1(x 1 )‖ [0,t)
)),γ ωb 2
1 (ηωa 2
2 (γω 1
2 (‖ω 1(x 1 )‖ [t0 ,t) ))),
˜σ 1 m (m),˜σω 1 (max { |ω 1 (x 1 (t 0 ))|,|ω 2 (x 2 (t 0 ))| } }
) , (D.104)