(ii) il existe $\phi \in K$ telle que pour tout $(x,y) \in \mathbb{R}^{2n}$, $|f(x) - f(y)| \le \phi(|x - y|)$ (on appelle $\phi$ le module de continuité de $f$ sur $\mathbb{R}^n$).

**Preuve.** (ii) ⇒ (i). Soit $\varepsilon \in \mathbb{R}_{>0}$, en définissant $\delta = \phi^{-1}(\varepsilon)$, on constate que (A.9) est bien vérifiée.

(i) ⇒ (ii). Définissons $\psi(s) = \sup_{|x-y| \le s} |f(x) - f(y)|$ pour $s \in \mathbb{R}_{\ge 0}$. On constate que $\psi(0) = 0$.

Montrons dans un premier temps que $\psi$ est bien définie sur $\mathbb{R}_{\ge 0}$. Soit $s \in \mathbb{R}_{\ge 0}$ et $(x,y) \in \mathbb{R}^{2n}$ tels que $|x - y| \le s$ et soit $\varepsilon \in \mathbb{R}_{>0}$ et $\delta \in \mathbb{R}_{>0}$ qui est défini tel que (A.9) est vérifiée. On définit $a_i = x + i \frac{y-x}{\delta}$ pour $i \in \{0, \dots, N\}$ avec $N = \lfloor \frac{y-x}{\delta} \rfloor$ où $\cdot$ représente la partie entière par défaut et $a_{N+1} = y$. Ainsi,

$$ 
\begin{aligned}
|f(x) - f(y)| &= \left| \sum_{i \in \{0, \dots, N\}} (f(a_i) - f(a_{i+1})) \right| \\
&\le \sum_{i \in \{0, \dots, N\}} |f(a_i) - f(a_{i+1})|,
\end{aligned} 
\quad (A.10) 
$$

puisque pour tout $i \in \{0, \dots, N\}$, $|a_i - a_{i+1}| \le \delta$, on a :

$$ |f(x) - f(y)| \le \sum_{i \in \{0, \dots, N\}} |f(a_i) - f(a_{i+1})| \le \sum_{i \in \{0, \dots, N\}} \varepsilon \le (N+1)\varepsilon, \quad (A.11) $$

par conséquent,

$$ |\psi(s)| \le (N+1)\varepsilon \quad (A.12) $$

et donc $\psi$ est bien définie sur $\mathbb{R}_{\ge 0}$. On remarque également que $\psi$ est croissante sur $\mathbb{R}_{\ge 0}$.

Nous allons maintenant montrer que $\psi$ est continue sur $\mathbb{R}_{\ge 0}$. Soit $a \in \mathbb{R}_{\ge 0}$, $\varepsilon \in \mathbb{R}_{>0}$, $\delta \in \mathbb{R}_{>0}$ tel que $\psi(\delta) \le \varepsilon$ et $s \in \mathbb{R}_{\ge 0}$ tel que $s = a + \eta$ avec $\eta \in [0, \delta]$, on a, puisque $\psi$ est croissante :

$$ 
\begin{aligned}
\psi(s) &= \psi(a + \eta) \\
&\le \psi(a + \delta) \\
&= \sup_{|x-y| \le a + \delta} |f(x) - f(y)| \\
&\le \sup_{(|x-z| \le a) \text{ et } (|z-y| \le \delta)} \left\{ |f(x) - f(z)| + |f(z) - f(y)| \right\} \quad (A.13) \\
&\le \sup_{|x-z| \le a} |f(x) - f(z)| + \sup_{|z-y| \le \delta} |f(z) - f(y)| \\
&= \psi(a) + \psi(\delta) \\
&\le \psi(a) + \varepsilon.
\end{aligned}
 $$

Par conséquent $\psi$ est continue à droite pour tout $a \in \mathbb{R}_{\ge 0}$. On peut montrer de la même manière que $\psi$ est continue à gauche pour tout $a \in \mathbb{R}_{\ge 0}$, ainsi $\psi$ est continue sur $\mathbb{R}_{\ge 0}$ tout entier. Seule la croissante stricte de $\psi$ nous manque pour pouvoir affirmer que $\psi \in K$. Pour cela, il suffit d'introduire la fonction $\phi : s \mapsto \psi(s) + s$. La propriété (ii) est ainsi vérifiée. $\square$
**Théorème A.2.1 (Théorème de la valeur moyenne [158])** Soient $a,b \in \mathbb{R}^n$, $f: \mathbb{R}^n \to \mathbb{R}^m$ dérivable sur $\mathbb{R}^n$ ($n,m \in \mathbb{Z}_{>0}$), alors il existe $c = a + (b-a)t$ où $t \in (0,1)$ tel que :

$$f(b) - f(a) = \partial f(c)(b-a). \qquad (A.14)$$

Le résultat suivant correspond au Lemme 3.4 dans [98].

**Lemme A.2.1 (Principe de comparaison)** Soit l'équation différentielle scalaire suivante :

$$\dot{y} = g(t,y), \quad y_0 \in \mathbb{R}$$

où $g$ est continue en $t$ et localement Lipschitzienne en $y$, pour tout $t \in [t_0, \infty)$ et $y \in J \subset \mathbb{R}$. Soit $t_{\max} \in (t_0, \infty]$ tel que $[t_0, t_{\max})$ est l'intervalle maximal d'existence des solutions $y(t)$ et supposons que $y(t) \in J$ pour tout $t \in [t_0, t_{\max})$. Soit $v$ une fonction dérivable qui satisfait l'inégalité suivante :

$$\dot{v}(t) \leq f(t,v(t)), \quad v_0 \leq y_0$$

et $v(t) \in J$ pour tout $t \in [t_0, t_{\max})$, alors, pour tout $t \in [t_0, t_{\max})$,

$$v(t) \leq u(t). \qquad (A.15)$$

Considérons le système suivant :

$$\begin{aligned} \dot{x} &= f(t,x,u) \\ y &= h(x), \end{aligned} \qquad (A.16)$$

où $x \in \mathbb{R}^n$, $y \in \mathbb{R}^p$, $u \in \mathbb{R}^m$, $f: \mathbb{R}_{\ge 0} \times \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ est localement Lipschitzienne et $h: \mathbb{R}^n \to \mathbb{R}^p$ est continue, $n,m,p \in \mathbb{Z}_{>0}$.

**Définition A.2.2 (Complétude positive)** Le système (A.16) est positivement complet si pour tout $(t_0, x_0, u) \in \mathbb{R}_{\ge 0} \times \mathbb{R}^n \times \mathcal{L}_{\infty}^m$, $x(\cdot, t_0, x_0, u)$ existe sur $[t_0, \infty)$.

**Remarque A.2.2** La complétude positive peut n'être définie que pour un ensemble limité de signaux d'entrée et de conditions initiales.

**Définition A.2.3** Le système (A.16) est uniformément positivement complet s'il existe $\chi_1, \chi_2$, $\chi_3 \in \mathcal{K}$ et $c \in \mathbb{R}_{>0}$ tels que, pour tout $(t_0, x_0, u) \in \mathbb{R}_{\ge 0} \times \mathbb{R}^n \times \mathcal{L}_{\infty}^m$ et $t \in [t_0, t_{\max})$ (qui représente l'intervalle maximal d'existence de solutions, $t_{\max} \in [t_0, \infty))$ :

$$|x(t,t_0,x_0,u)| \leq \chi_1(t-t_0) + \chi_2(|x_0|) + \chi_3(\|u\|_{[t_0,t)}) + c, \qquad (A.17)$$

ainsi $t_{\max} = \infty$.

Cette définition est inspirée de [6] où il est prouvé que, si le système (A.16) est à temps invariant, complétude positive et (A.17) (avec $\chi_1 = 0$) sont équivalents (voir Corollaire 2.3). La notion de complétude positive uniforme est également utilisée dans [122] (Définition 5).
# Annexe B

## Rappels de stabilité

### B.1 Stabilité des systèmes à temps variant

Soit le système dont les dynamiques sont représentées par l'équation différentielle ordinaire suivante :

$$ \dot{x} = f(t,x), \qquad (B.1) $$

où $x \in \mathbb{R}^n$, $f: \mathbb{R}_{\ge 0} \times \mathbb{R}^n \to \mathbb{R}^n$ telle que $f(t,0) = 0$ pour tout $t \in \mathbb{R}_{\ge 0}$, $n \in \mathbb{Z}_{>0}$.

**Définition B.1.1 (UGAS)** On dit que l'origine est uniformément globalement asymptotiquement stable (UGAS) pour le système (B.1) s'il existe $\beta \in K\mathcal{L}$ telle que, pour tout $x_0 \in \mathbb{R}^n$, les solutions de (B.1) vérifient :

$$ |x(t)| \le \beta(|x_0|, t - t_0) \quad \forall t \ge t_0 \ge 0. \qquad (B.2) $$

Si le système (B.1) est autonome, on dit que 0 est GAS (l'uniformité est superflue).

Il est généralement difficile de prouver la stabilité de l'origine de (B.1) à partir de la Définition B.1.1. Les fonctions de Lyapunov offre un moyen pratique d'y arriver. Le théorème suivant se déduit du Théorème 4.9 et du Lemme 4.3 dans [98].

**Théorème B.1.1** L'origine est UGAS pour (B.1) s'il existe une fonction $V \in C^1(\mathbb{R}_{\ge 0} \times \mathbb{R}^n, \mathbb{R})$, $\alpha_1, \alpha_2 \in K_\infty$ et $\alpha_3 \in K$ tels que,

(i) pour tout $(t,x) \in \mathbb{R}_{\ge 0} \times \mathbb{R}^n$, $\alpha_1(|x|) \le V(t,x) \le \alpha_2(|x|)$;

(ii) le long des solutions de (B.1), $\dot{V}(t,x) = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x}f(t,x) \le -\alpha_3(|x|)$.

On dit dans ce cas que $V$ est une fonction de Lyapunov (forte) pour le système (B.1).

**Remarque B.1.1** La condition (i) est équivalente au fait que $V$ est définie positive et radialement non bornée et l'inégalité de (ii) revient à dire que $\dot{V}$ est définie négative (cf. Lemme 4.3 dans [98]).
Il est souvent difficile de trouver des fonctions de Lyapunov fortes. Parfois, la dérivée de la fonction considérée est seulement définie semi-négative ($\dot{V} \le 0$), on parle alors de fonction de Lyapunov faible. Tout ce que l'on peut conclure, a priori, est que l'origine du système est stable, i.e. il existe $\alpha \in \mathcal{K}$ telle que pour tout $t_0 \in \mathbb{R}_{\ge 0}$, $x_0 \in \mathbb{R}^n$, $|x(t)| \le \alpha(|x_0|)$. Des outils d'analyse sont disponibles afin de déduire de plus fortes propriétés à partir de fonctions de Lyapunov faibles, comme le principe d'invariance de LaSalle. Le résultat suivant correspond au Théorème 4.4 dans [98].

**Théorème B.1.2 (Principe d'invariance de Krasovskii-LaSalle)** Soient $D \subset \mathbb{R}^n$ et

- $\Omega \subset D$ un ensemble compact positivement invariant pour le système (B.1);

- $V \in C^1(D; \mathbb{R})$ telle que $\dot{V}(x) \le 0$ sur $\Omega$;

- E l'ensemble des points de $\Omega$ où $\dot{V}(x) = 0$;

- M le plus grand invariant inclus dans E.

Alors toute solution de (B.1) telle que $x(0) \in \Omega$ converge asymptotiquement vers M.

Ce corollaire est une conséquence directe du principe d'invariance de LaSalle-Krasovskii (voir Corollaires 4.1 et 4.2 dans [98]).

**Corollaire B.1.1** Soient $D \subset \mathbb{R}^n$ tel que $0 \in D$ et

- $V \in C^1(D; \mathbb{R})$ une fonction définie positive sur D, telle que $\dot{V}(x) \le 0$ sur D;

- $S = \{x \in D : \dot{V}(x) = 0\}$ et supposons qu'aucune solution ne puisse rester dans S identiquement, à part $x = 0$;

Alors $x=0$ est asymptotiquement stable. Si $D = \mathbb{R}^n$ et $V$ est radialement non bornée, $x=0$ est globalement asymptotiquement stable.

## B.2 Stabilité des systèmes munis d'entrées et de sorties

Considérons le système suivant :

$$ 
\begin{aligned}
\dot{x} &= f(t,x,u) \\
y &= h(x),
\end{aligned} 
\qquad (B.3) 
$$

où $x \in \mathbb{R}^n$, $u \in \mathbb{R}^m$, $y \in \mathbb{R}^p$, $f : \mathbb{R}_{\ge 0} \times \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ telle que $f(t,0,0) = 0$ pour tout $t \in \mathbb{R}_{\ge 0}$ et $h : \mathbb{R}^n \times \mathbb{R}^p, n,m,p \in \mathbb{Z}_{>0}$.

**Définition B.2.1 (UEBEB / UGS)** Le système (B.3) est uniformément-entrée-bornée-état-borné (UEBEB) avec $u$ comme entrée, s'il existe $\alpha, \gamma \in \mathcal{K}$ tels que, pour tout $x_0 \in \mathbb{R}^{n_x}$, $u \in \mathcal{L}_{\infty}^{n_u}$:

$$ |x(t)| \le \alpha(|x_0|) + \gamma(\|u\|_{\infty}) \quad \forall t \ge t_0 \ge 0, \qquad (B.4) $$

et si $\gamma$ est une fonction linéaire, le système (B.3) est dit UEBEB avec un gain linéaire. Lorsqu'aucune entrée n'affecte les dynamiques du système (i.e. $u = 0$), on dit que celui-ci est uniformément globalement stable (UGS).
**Définition B.2.2 (Stabilité entrée-sortie / entrée-état)** On dit que le système (B.3) est stable entrée-sortie de l'entrée u vers la sortie y s'il existe $\beta \in KL$ et un gain $\gamma \in K$, tels que pour tout $x_0 \in \mathbb{R}^{n_x}$, $u \in \mathcal{L}_{\infty}^{n_u}$:

$$|y(t)| \leq \beta(|x_0|, t - t_0) + \gamma(\|u\|_{\infty}) \quad \forall t \geq t_0 \geq 0. \qquad (B.5)$$

Lorsque (B.5) est vérifiée pour $y = x$, le système (B.3) est stable entrée-état vis-à-vis de $u$. Si $\gamma$ est une fonction linéaire, on dit que le système (B.3) est stable entrée-sortie/entrée-état avec un gain linéaire.

Différentes caractérisations des stabilités entrée-sortie et entrée-état sont listées.

**Proposition B.2.1** Le système (B.3) est stable entrée-sortie de l'entrée u vers la sortie y si et seulement si l'une des propriétés suivantes est vérifiée:

(i) il existe $\beta \in KL$ et un gain $\gamma \in K$ tels que :

$$|y(t)| \leq \max\{\beta(|x_0|, t - t_0), \gamma(\|u\|_{\infty})\} \quad \forall t \geq t_0 \geq 0; \qquad (B.6)$$

(ii) (si le système (B.3) est UEBEB) il existe $V \in C^{\infty}(\mathbb{R}^n, \mathbb{R}_{\geq 0})$, $\alpha_1, \alpha_2 \in K_{\infty}$, $\chi \in K$ et $\alpha_3 \in KL$ tels que :

- pour tout $x \in \mathbb{R}^n$, $\alpha_1(|h(x)|) \leq V(x) \leq \alpha_2(|h(x)|)$;

- pour tout $(x,u) \in \mathbb{R}^{n+m}$, $V(x) \geq \chi(|u|) \Rightarrow \frac{\partial V}{\partial x}f(x,u) \leq -\alpha_3(V(x),|x|)$.

**Preuve.** La propriété (i) est bien équivalente à la Définition B.2.2 d'après la Proposition A.1.2. L'équivalence avec (ii) est obtenue à l'aide du Théorème 1 dans [186].

La proposition suivante est démontrée dans [185] (voir Théorème 1).

**Proposition B.2.2** Le système (B.3) est stable entrée-état vis-à-vis de $u$ si et seulement si l'une des propriétés suivantes est vérifiée:

(i) il existe $\beta \in KL$ et un gain $\gamma \in K$ tels que :

$$|x(t)| \leq \max\{\beta(|x_0|, t - t_0), \gamma(\|u\|_{\infty})\} \quad \forall t \geq t_0 \geq 0; \qquad (B.7)$$

(ii) il existe $V \in C^{\infty}(\mathbb{R}^n, \mathbb{R}_{\geq 0})$, $\alpha_1, \alpha_2 \in K_{\infty}$, $\chi \in K$ et $\alpha_3 \in K$ tels que :

- pour tout $x \in \mathbb{R}^n$, $\alpha_1(|x|) \leq V(x) \leq \alpha_2(|x|)$;

- pour tout $(x,u) \in \mathbb{R}^{n+m}$, $\chi(|x|) \geq |u| \Rightarrow \frac{\partial V}{\partial x}f(x,u) \leq -\alpha_3(|x|)$.

**Remarque B.2.1** Lorsque la stabilité entrée-état est prouvée à l'aide de la condition (ii), on peut prendre $\gamma = \alpha_1^{-1} \circ \alpha_2 \circ \chi^{-1}$ dans (i).
# Annexe C

## Preuve de la Proposition 4.3.1

La preuve est fortement inspirée de celle du Lemme 11 dans [165]. On peut montrer, d'après (4.37), que pour tout $\mu \in \mathbb{R}_{>0}$, $i \in \mathbb{Z}_{\ge 0}$ et $e \in \mathbb{R}^{n_e}$, avec $\tilde{\alpha}_3 = \frac{2}{\mu}\alpha_3$:

$$ \frac{2}{\mu} [W(i+1,h(i,e)) - W(i,e)] \leq -\tilde{\alpha}_3(W(i,e)). \quad (C.1) $$

Considérons $a \in \mathcal{K} \cap C^1(\mathbb{R}_{\ge 0}, \mathbb{R}_{\ge 0})$ telle que, pour tout $\tau \in \mathbb{R}_{\ge 0}$:

$$ a(\tau) \leq \min\{\tau, \tilde{\alpha}_3(\tau)\} \quad \text{et} \quad a'(\tau) < 1. \quad (C.2) $$

Un exemple de fonction $a$ est donné par l'équation (140) dans [165] : $a(\tau) = \frac{2}{\pi} \int_0^\tau \frac{\min\{s, \tilde{\alpha}_3(s)\}}{1+s^2} ds$ qui satisfait bien les conditions requises. En effet, on peut prouver que $a \in \mathcal{K} \cap C^1(\mathbb{R}_{\ge 0}, \mathbb{R}_{\ge 0})$ et $a(\tau) \le \min\{\tau, \tilde{\alpha}_3(\tau)\}$, de plus $a'(\tau) = \frac{2}{\pi} \frac{\min\{\tau, \tilde{\alpha}_3(\tau)\}}{1+\tau^2} \le \frac{2}{\pi} \frac{\tau}{1+\tau^2} \le \frac{1}{\pi} < 1$, pour $\tau \in \mathbb{R}_{\ge 0}$.

Soit

$$ \rho(\tau) = \begin{cases} \exp\left(\int_1^\tau \frac{2}{a(s)} ds\right) & \text{pour } \tau > 0 \\ 0 & \text{pour } \tau = 0. \end{cases} \quad (C.3) $$

Cette fonction est continue et croissante sur $\mathbb{R}_{\ge 0}$. De plus, puisque $s \mapsto \int_1^\tau \frac{2}{a(s)} ds$ diverge en $-\infty$ quand $\tau$ tend vers 0 et vers $+\infty$ quand $\tau$ tend vers $+\infty$, $\rho \in \mathcal{K}_\infty$. Montrons que $\rho \in C^1(\mathbb{R}_{\ge 0}, \mathbb{R}_{\ge 0})$. Puisque $\rho \in C^1(\mathbb{R}_{>0}, \mathbb{R}_{\ge 0})$, il suffit de prouver que:

$$ \rho'(0) = 0 \quad \text{et} \quad \lim_{\tau \to 0^+} \rho'(\tau) = 0. \quad (C.4) $$

Au voisinage de 0,

$$ \rho(\tau) = \exp\left(-\int_{\tau}^{1} \frac{2}{a(s)} ds\right) \le \exp\left(-\int_{\tau}^{1} \frac{2}{s} ds\right) = \exp(\ln \tau^2) = \tau^2. \quad (C.5) $$

Par conséquent, $\rho'(0)$ existe et est égal à 0. D'autre part, pour $\tau \in \mathbb{R}_{>0}$:

$$ \rho'(\tau) = \frac{2}{a(\tau)} \rho(\tau) \quad \text{et} \quad \rho''(\tau) = 2(2 - a'(\tau)) \frac{\rho(\tau)}{a^2(\tau)}. \quad (C.6) $$

Puisque $a'(\tau) < 2$, $\rho''$ est strictement positive sur $\mathbb{R}_{>0}$ et $\rho'$ est donc strictement croissante sur cet espace. Ainsi, sachant que $\rho'$ est également positive, $\lim_{\tau \to 0^+} \rho'(\tau)$ existe et est positive
ou nulle. Raisonnons par l'absurde et supposons que $\lim_{\tau \to 0^+} \rho'(\tau) > 0$. Il existe donc $c, \delta \in \mathbb{R}_{>0}$ tels que, pour tout $\tau \in (0,\delta)$,

$$ \rho'(\tau) \geq c, \qquad (C.7) $$

ce qui implique, sachant $\rho''(\tau) \geq \frac{\rho'(\tau)}{a(\tau)}$ d'après (C.6) et le fait que $a'(\tau) < 1$, que :

$$ \rho''(\tau) \geq \frac{c}{a(\tau)} \geq \frac{c}{\tau}. \qquad (C.8) $$

Par conséquent, puisque

$$ \rho'(\tau) = \rho'(\delta) - \int_{\tau}^{\delta} \rho''(s) \, ds, \qquad (C.9) $$

$\lim_{\tau \to 0^+} \rho'(\tau) = -\infty$, ce qui est impossible, par conséquent $\lim_{\tau \to 0^+} \rho'(\tau) = 0$ et $\rho \in C^1(\mathbb{R}_{\ge 0}, \mathbb{R}_{\ge 0})$.

Considérons maintenant $U = \rho \circ W$. Pour tout $i \in \mathbb{Z}_{\ge 0}$ et $e \in \mathbb{R}^{n_e}$, puisque $\rho \in C^1(\mathbb{R}_{\ge 0}, \mathbb{R}_{\ge 0})$, d'après le théorème de la valeur moyenne, il existe $W^* \in [W(i+1,h(i,e)), W(i,e)]$ ($W(i+1,h(i,e)) \le W(i,e)$) :

$$ U(i+1,h(i,e)) - U(i,e) = \rho'(W^*)[W(i+1,h(i,e)) - W(i,e)], \qquad (C.10) $$

en invoquant (C.1) et sachant que $\rho'$ est positive,

$$ U(i+1,h(i,e)) - U(i,e) \leq -\frac{\mu}{2}\rho'(W^*)\tilde{\alpha}_3(W(i,e)), \qquad (C.11) $$

sachant que $\rho'$ est également croissante sur $\mathbb{R}_{\ge 0}$ et que $W(i+1,h(i,e)) \le W^* \le W(i,e)$,

$$ U(i+1,h(i,e)) - U(i,e) \leq -\frac{\mu}{2}\rho'(W(i+1,h(i,e)))\tilde{\alpha}_3(W(i+1,h(i,e))), \qquad (C.12) $$

puisque $\rho'(\tau) = \frac{2}{a(\tau)}\rho(\tau)$ pour $\tau \in \mathbb{R}_{>0}$ et $\rho'(0) = 0$,

$$ U(i+1,h(i,e)) - U(i,e) \leq -\frac{\mu}{a(W(i+1,h(i,e)))}U(i+1,h(i,e))\tilde{\alpha}_3(W(i+1,h(i,e))), $$

d'après la définition de $a$, $a(\tau) \le \tilde{\alpha}_3(\tau)$ pour $\tau \in \mathbb{R}_{\ge 0}$, donc

$$ U(i+1,h(i,e)) - U(i,e) \leq -\mu U(i+1,h(i,e)). $$

En conséquence, pour $\lambda = \frac{1}{1+\mu} \in (0,1)$,

$$ U(i+1,h(i,e)) \leq \lambda U(i,e). $$

La preuve se conclut en remarquant que, puisque $\rho \in C^1(\mathbb{R}_{\ge 0}, \mathbb{R}_{\ge 0})$, $U$ hérite des propriétés de continuité de $W$. $\square$
# Annexe D

## Théorèmes du petit gain pour des systèmes paramétrés

Dans cette annexe, des théorèmes du petit gain sont développés à partir des travaux de [184]. Deux types d'interconnexions sont étudiés : celles dites « classiques » entre deux sous-systèmes vérifiant des propriétés de type stabilité entrée-état, d'autres plus atypiques puisque trois sous-systèmes sont interconnectés, l'un d'eux ne satisfaisant que des propriétés de type entrée-bornée-état-borné. De plus, pour la première fois à notre connaissance des théorèmes du petit gain sont présentés pour des systèmes paramétrés en temps continu et explicitement prouvés (un théorème est bien proposé sous des conditions différentes des nôtres dans [191] mais sans sa preuve). A noter que l'interconnexion de systèmes paramétrés discrets est analysé à l'aide d'un théorème du petit gain dans [111] basé sur la construction d'une fonction de Lyapunov pour le système global.

Ces résultats sont utilisés dans ce mémoire pour analyser la stabilité des observateurs émulés pour les NCS. Leur portée s'avère toutefois plus générale. Nous montrons ainsi dans [163] qu'ils peuvent être appliqués pour la stabilisation semiglobale pratique asymptotique de classes de systèmes non linéaires ainsi que pour la commande et la poursuite de trajectoires pour les systèmes à données échantillonnées.

Cette annexe est organisée de la façon suivante. Des définitions et résultats préliminaires sont présentés dans §D.1, fournissant des versions locales de certaines propriétés de [121, 184]. Des théorèmes du petit gain sont ensuite développés et prouvés dans §D.2, puis étendus aux systèmes paramétrés dans §D.3.
## D.1 Définitions et résultats préliminaires

Considérons le système :

$$ \dot{x} = f(t,x,u) \quad \forall t \in [t_{i-1}, t_i] \qquad (D.1) $$

$$ y = h(t,x,u) \qquad (D.2) $$

$$ x(t_i^+) = g(t_i, x(t_i), u(t_i)), \qquad (D.3) $$

où $x \in \mathbb{R}^{n_x}$ est l'état, $y \in \mathbb{R}^{n_y}$ la sortie, $u \in \mathbb{R}^{n_u}$ l'entrée, $f : \mathbb{R}_{\ge 0} \times \mathbb{R}^{n_x+n_u} \to \mathbb{R}^{n_x}$ est continue en $x$ et $u$ et continue par morceaux en $t$, $h : \mathbb{R}_{\ge 0} \times \mathbb{R}^{n_x+n_u} \to \mathbb{R}^{n_y}$ est continue, $n_x, n_u, n_y \in \mathbb{Z}_{>0}$. La séquence de sauts $\{t_i\}_{i \in \mathbb{Z}_{>0}}$ vérifie :

$$ v \le t_{i+1} - t_i \le \tau, \qquad (D.4) $$

avec $\tau \in [v, \infty)$ et $t_0 \in \mathbb{R}_{\ge 0}$ est l'instant initial.

**Remarque D.1.1** Nous avons choisi de considérer une classe de systèmes hybrides qui comprend celles étudiées dans la Partie II. Les résultats de cette annexe s’appliquent directement aux systèmes dont les dynamiques sont uniquement continues [163].

Cette section a pour but d’obtenir un résultat technique (Proposition D.1.3) qui nous permettra de conclure dans la preuve des théorèmes du petit gain. Une série de définitions, lemmes et propositions nécessaires à l’obtention dudit résultat sont maintenant énoncés.

**Définition D.1.1** Soient $\mu \in \mathcal{K}$, $c, \Delta \in \mathbb{R}_{>0}$, le système (D.1)-(D.3) est dit $(\mu, c, \Delta)$-uniformément pratiquement stable entrée-sortie<fnref n="1" /> si les conditions suivantes sont vérifiées :

(i) [stabilité $(\mu, c, \Delta)$-uniforme pratique] il existe $\delta \in \mathcal{K}_\infty$ telle que, pour tout $\varepsilon \in (0, \delta^{-1}(\Delta))$, $x(t_0) \in \mathbb{R}^{n_x}$ et $u \in \mathcal{L}_\infty^{n_u}$ avec $\max \{|x(t_0)|, \|u\|_\infty\} < \delta(\varepsilon)$ : \\
$$ |y(t)| \le \max \{\varepsilon, \mu(\|u\|_{[t_0, t)}), c\} \quad \forall t \ge t_0 \ge 0, \qquad (D.5) $$
lorsque $\Delta = \infty$, le système (D.1)-(D.3) est $(\mu, c)$-uniformément pratiquement stable;

(ii) [attractivité $(\mu, c, \Delta)$-uniforme pratique] pour tout $r \in (0, \Delta)$, $\varepsilon \in (c, \infty)$, il existe $T(r, \varepsilon) \in \mathbb{R}_{>0}$ tel que pour tout $x(t_0) \in \mathbb{R}^{n_x}$ et $u \in \mathcal{L}_\infty^{n_u}$ avec $\max \{|x(t_0)|, \|u\|_\infty\} < r$, pour tout $t_0 \in \mathbb{R}_{\ge 0}$ : \\
$$ |y(t)| \le \max \{\varepsilon, \mu(\|u\|_{[t_0, t)})\} \quad \forall t \in [t_0 + T(\varepsilon, r), \infty), \qquad (D.6) $$
lorsque $\Delta = \infty$, le système (D.1)-(D.3) est $(\mu, c)$-uniformément pratiquement attractif.

Le lemme ci-dessous découle directement du Lemme 3.1 dans [121].

**Lemme D.1.1** Soient $\Delta \in \mathbb{R}_{>0}$ et $\Phi : [0, \Delta) \times [t_0, \infty) \to \mathbb{R}_{\ge 0}$ tels que, pour tout $r \in (0, \Delta)$, $\varepsilon \in \mathbb{R}_{>0}$, il existe $\bar{T}(r, \varepsilon)$ tel que, pour tout $s \in [0, r)$, $t \ge t_0 + \bar{T}(r, \varepsilon)$, $\Phi(s, t) < \varepsilon$. Alors, il existe $T : (0, \Delta) \times \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ qui satisfait les propriétés suivantes :

(i) pour tout $r \in (0, \Delta)$, $T(r, \cdot) : \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ est continue, strictement décroissante et bijective de $\mathbb{R}_{>0}$ dans $\mathbb{R}_{>0}$;

<footnote n="1">1. L'entrée et la sortie sont respectivement $u$ et $y$ dans cette section.</footnote>
(ii) pour tout $\varepsilon \in \mathbb{R}_{>0}$, $T(\cdot, \varepsilon)$ est strictement croissante sur $(0,\Delta)$;
telle que pour tout $r \in (0,\Delta)$, $\varepsilon \in \mathbb{R}_{>0}$, $s \in [0,r)$ et $t \ge t_0 + T(r,\varepsilon)$, $\Phi(s,t) < \varepsilon$.

**Preuve abrégée.** La preuve est la même que celle du Lemme 3.1 dans [121]. En effet, en définissant, pour $r \in (0,\Delta)$ et $\varepsilon \in \mathbb{R}_{>0}$, l'ensemble :

$$ \mathcal{A}(r,\varepsilon) = \{T \in \mathbb{R}_{\ge 0} : \forall s \in [0,r) \ \forall t \ge t_0 + T \ \Phi(s,t) < \varepsilon\}, \quad (D.7) $$

et $\bar{T}(r,\varepsilon) = \inf \{\mathcal{A}(r,\varepsilon)\}$, $T(r,\varepsilon) = \frac{2}{\varepsilon} \int_{\frac{\varepsilon}{2}}^{\varepsilon} \bar{T}(r,s) \, ds + \frac{r}{\varepsilon}$, les conditions (i) et (ii) sont satisfaites, en invoquant les mêmes arguments que dans [121]. $\square$

La proposition suivante est une variation de la Proposition A.1 dans [184].

**Proposition D.1.1** Soient $\Delta \in \mathbb{R}_{>0}$ et $\Phi : [0,\Delta) \times [t_0,\infty) \to \mathbb{R}_{\ge 0}$ tels que :

(i) pour tout $r \in (0,\Delta)$, $\varepsilon \in \mathbb{R}_{>0}$, il existe $\bar{T}(r,\varepsilon)$ tel que, pour tout $s \in [0,r)$ et $t \ge t_0+\bar{T}(r,\varepsilon)$, $\Phi(s,t) < \varepsilon$;

(ii) il existe $\delta \in \mathcal{K}_{\infty}$ tel que pour tout $\varepsilon \in (0,\delta^{-1}(\Delta))$, $s \in [0,\delta(\varepsilon))$ et $t \ge t_0$, $\Phi(s,t) < \varepsilon$; alors, il existe $\beta \in \mathcal{KL}$ tel que, pour $s \in [0,\Delta)$ et $t \ge t_0$, $\Phi(s,t) < \beta(s,t - t_0)$.

**Preuve.** La preuve suit les mêmes lignes que la seconde partie de la preuve de la Proposition 2.5 dans [121]. Soit $\Delta \in \mathbb{R}_{>0}$ et définissons $\varphi = \delta^{-1} \in \mathcal{K}_{\infty}$. Ainsi, d'après (ii), pour tout $s \in [0,\Delta)$ et $t \in [t_0,\infty)$,

$$ \Phi(s,t) < \delta^{-1}(s) = \varphi(s). \quad (D.8) $$

Puisque la condition (i) est vérifiée, le Lemme D.1.1 est utilisé pour construire la fonction $T$. Soit $\psi(r,\cdot) = T^{-1}(r,\cdot)$ pour tout $r \in (0,\Delta)$, qui hérite des propriétés suivantes de $T$ :

- pour tout $r \in (0,\Delta)$, $\psi(r,\cdot) : \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ est continue, bijective de $\mathbb{R}_{>0}$ dans $\mathbb{R}_{>0}$, et strictement décroissante, de plus, on définit $\psi(r,0) = \infty$ en accord avec le fait que $\lim_{t \to 0^+} \psi(r,t) = \infty$;

- $\psi(\cdot,t)$ est strictement croissante sur $(0,\Delta)$ pour tout $t \in \mathbb{R}_{>0}$.

**Fait D.1.1** Pour tout $r \in (0,\Delta)$, $s \in [0,r)$, $t \ge t_0$, $\Phi(s,t) \le \psi(r,t - t_0)$.

**Preuve du fait.** La preuve est la même que dans [121], mais nous avons choisi de la présenter avec nos notations. De par la définition de $T$, pour tout $r \in (0,\Delta)$, $\varepsilon \in \mathbb{R}_{>0}$, $s \in [0,r)$ et $t \ge t_0 + T(r,\varepsilon)$ :

$$ \Phi(s,t) < \varepsilon. \quad (D.9) $$

En remarquant que $t - t_0 = T(r,\psi(r,t - t_0))$ si $t - t_0 > 0$, et que $\psi(r,\cdot)$ est une bijection de $\mathbb{R}_{>0}$ dans $\mathbb{R}_{>0}$, il s'en suit d'après (D.9) que, pour tout $r \in (0,\Delta)$ et $s \in [0,r)$, $t \in (t_0,\infty)$ :

$$ \Phi(s,t) < \psi(r,t - t_0). \quad (D.10) $$
En notant que $\psi(r,0) = \infty$, l'inégalité désirée est alors assurée pour $t = t_0$. $\square$

Soit $\tilde{\psi}(r,t) = \min\{\varphi(r), \psi(r,t)\}$ pour $r \in (0,\Delta)$ et $t \in \mathbb{R}_{\ge 0}$. On remarque que $\tilde{\psi}$ est décroissante en $t$ et $\lim_{t \to \infty} \tilde{\psi}(s,t) = 0$ ($\lim_{t \to \infty} \psi(s,t) = 0$ puisque $\psi(r,\cdot)$ est une bijection strictement décroissante de $\mathbb{R}_{>0}$ dans $\mathbb{R}_{\ge 0}$). On obtient, d'après (D.8) et le Fait D.1.1, que pour tout $r \in (0,\Delta)$, $s \in [0,r)$, $t \ge t_0$:

$$ \Phi(s,t) \le \tilde{\psi}(r,t - t_0), \qquad (D.11) $$

par conséquent, d'après (D.8), (D.11) et en prolongeant par continuité $\tilde{\psi}$ sur $[0,\Delta] \times [t_0,\infty)$ par $\tilde{\psi}(\Delta, t - t_0) = \lim_{r \to \Delta^{-}} \tilde{\psi}(r, t - t_0) \in \mathbb{R}_{>0}$, (si $\psi(\cdot, t)$ tend vers l'infini en $\Delta$, $\varphi$ prend bien une valeur constante strictement positive puisque elle est de classe $\mathcal{K}_{\infty}$) et $\tilde{\psi}(0, t - t_0) = 0$ (puisque $\psi$ est positive et que $\varphi(0) = 0$) on a :

$$ \Phi(s,t) \le \sqrt{\phi(s)} \sqrt{\tilde{\psi}(r,t - t_0)} \le \sqrt{\phi(s)} \sqrt{\tilde{\psi}(\Delta, t - t_0)}. \qquad (D.12) $$

Soit $\beta(s,t) = \sqrt{\phi(s)} \sqrt{\tilde{\psi}(\Delta,t)}$, pour $s \in [0,\Delta)$ et $t \in \mathbb{R}_{\ge 0}$, on peut voir que :

- pour tout $s \in [0,\Delta)$, $\beta(s,\cdot)$ est continue, décroissante et $\lim_{t \to \infty} \beta(s,t) = 0$ (comme $\tilde{\psi}(s,\cdot)$ est décroissante et $\lim_{t \to \infty} \tilde{\psi}(s,t) = 0$);

- pour tout $t \in \mathbb{R}_{\ge 0}$, $\beta(\cdot, t)$ est continue, strictement croissante sur $[0,\Delta)$ et s'annule en l'origine.

De cette manière, $\beta \in \mathcal{KL}$, ce qui clôt la preuve. $\square$

Le résultat suivant provient du Lemma A.2 dans [184].

**Lemme D.1.2** Soient $\Delta, c \in \mathbb{R}_{>0}$, $\delta \in \mathcal{K}_{\infty}$, $c \in \mathbb{R}_{>0}$ et $T : (0,\Delta) \times (c,\infty) \to \mathbb{R}_{\ge 0}$ qui satisfait :

(i) $T(\cdot, \varepsilon)$ est croissante pour tout $\varepsilon \in (c, \infty)$;

(ii) $\lim_{\varepsilon \to c^{-}} T(r, \varepsilon) = \infty$, pour tout $r \in (0, \Delta)$;

alors, pour tout $\bar{\eta} > 1$, il existe $\beta \in \mathcal{KL}$ tel que pour tout $r \in (0,\Delta)$, $t \in \mathbb{R}_{\ge 0}$, il existe $\varepsilon \in A_T(r,t) = \{\eta \in (c,\infty) : T(r,\eta) \le t\} \cup \{\infty\}$ tel que

$$ \min \{\varepsilon - \bar{\eta}c, \delta^{-1}(r)\} \le \beta(r,t). \qquad (D.13) $$

**Preuve abrégée.** La preuve se déduit de celle du Lemme A.2 dans [184]. Soit $\bar{\eta} > 1$. On peut montrer que pour tout $r \in (0,\Delta)$ et $t \in \mathbb{R}_{\ge 0}$, $\inf\{A_T(r,t)\} > c$, la fonction suivante est définie, pour $r \in (0,\Delta)$ et $t \in \mathbb{R}_{\ge 0}$,

$$ \varphi(r,t) = \min \{ \inf\{A_T(r,t)\} - c, \delta^{-1}(r) \}, \qquad (D.14) $$

avec $\varphi(0,t) = 0$ et $\varphi(r,t) = \delta^{-1}(r)$ tant que $\inf\{A_T(r,t)\} = \infty$. On peut alors montrer que $\varphi$ vérifie les conditions (i) et (ii) de la Proposition D.1.1. Ainsi il existe $\bar{\beta} \in \mathcal{KL}$ tel que, pour tout $r \in (0,\Delta)$, $t \in \mathbb{R}_{\ge 0}$:

$$ \varphi(r,t) \le \bar{\beta}(r,t). \qquad (D.15) $$
Soit $\beta = \bar{\eta}\bar{\beta}$, pour $r \in (0,\Delta)$ et $t \in \mathbb{R}_{\ge 0}$, $\inf\{A_T(r,t)\} \in (0,\infty]$.

Si $\min\{\inf\{A_T(r,t)\} - c, \delta^{-1}(r)\} = \inf\{A_T(r,t)\}$, alors il existe $\varepsilon \in A_T(r,t)$ tel que $\varepsilon < \bar{\eta} \inf\{A_T(r,t)\}$. D'après le fait suivant

$$ \min \{ \inf\{A_T(r,t)\} - c, \delta^{-1}(r) \} \le \bar{\beta}(r,t), \qquad (D.16) $$

et puisque $\varepsilon - \bar{\eta}c < \bar{\eta}\inf\{A_T(r,t)\} - \bar{\eta}c$, (D.13) est obtenu. D'autre part, si $\min\{\inf\{A_T(r,t)\} - c, \delta^{-1}(r)\} = \delta^{-1}(r)$, puisque $\delta^{-1}(r) \le \bar{\beta}(r,t)$, (D.13) est vérifiée. $\square$

La proposition suivante offre une caractérisation de la stabilité entrée-sortie $(\mu,c,\Delta)$-uniforme et pratique à l'aide de fonctions de comparaison.

**Proposition D.1.2** Si le système (D.1)-(D.3) est $(\mu,c,\Delta)$-uniformément pratiquement stable entrée-sortie, alors, pour tout $\eta > 1$, il existe $\beta \in \mathcal{KL}$ tel que pour tout $x(t_0) \in \mathbb{R}^{n_x}$ et $u \in \mathcal{L}_{\infty}^{n_u}$ avec $\max\{|x(t_0)|, \|u\|_{\infty}\} < \Delta$,

$$ |y(t)| \le \max \{\beta(|x(t_0)|, t - t_0), \mu(\|u\|_{[t_0,t)}), \eta c\} \quad \forall t \ge t_0 \ge 0. \qquad (D.17) $$

**Preuve abrégée.** La preuve suit celle de la Proposition 2.7 dans [184]. Soit $\eta > 1$, $\bar{\eta} \in (1,\eta)$, définissons $\nu = \frac{\eta}{\bar{\eta}} - 1 \in \mathbb{R}_{>0}$. Soit $t_0 \in \mathbb{R}_{\ge 0}$, $r = |x(t_0)| < \Delta$, $u \in \mathcal{L}_{\infty}^{n_u}$ avec $\|u\|_{\infty} < \Delta$, puisque le système (D.1)-(D.3) est $(\mu,c,\Delta)$-uniformément pratiquement stable, on a, pour tout $t \in [t_0,\infty)$,

$$ |y(t)| \le \max \{\delta^{-1}(r), \mu(\|u\|_{[t_0,t)}), c\}, \qquad (D.18) $$

d'autre part, puisque le système (D.1)-(D.3) est $(\mu,c,\Delta)$-uniformément pratiquement attractif, il existe $\bar{T} : (0,\Delta) \times (c,\infty) \to [t_0,\infty)$ tel que pour tout $\varepsilon \in (c,\infty)$, $t \ge t_0 + \bar{T}(r,\varepsilon)$,

$$ |y(t)| \le \max \{\varepsilon, \mu(\|u\|_{[t_0,t)})\}. \qquad (D.19) $$

Soit, pour $(r,\varepsilon) \in (0,\Delta) \times (c,\infty)$,

$$ T(r,\varepsilon) = \frac{r}{\varepsilon - c} + \inf \{ \bar{T}(r',\varepsilon') : r' \ge r \text{ et } \varepsilon' \in (c,\varepsilon) \} \qquad (D.20) $$

qui est croissante en $r$ pour tout $\varepsilon \in (c,\infty)$ et $\lim_{\varepsilon \to c^-} T(r,\varepsilon) = \infty$ pour tout $r \in (0,\Delta)$ qui, ainsi, satisfait les conditions (i) et (ii) du Lemme D.1.2. De cette manière, soit $t \in [t_0,\infty)$ et prenons $\varepsilon \in A_T(r,t - t_0)$, où $A_T(r,t - t_0)$ est défini au Lemme D.1.2, on a :

$$ |y(t)| \le \max \{\varepsilon, \mu(\|u\|_{[t_0,t)})\}, \qquad (D.21) $$

d'après le Lemme D.1.2,

$$ \min \{\varepsilon - \bar{\eta}c, \delta^{-1}(r)\} \le \beta(r, t - t_0). \qquad (D.22) $$

Dans le cas où $\varepsilon - \bar{\eta}c < \delta^{-1}(r)$, nécessairement $\varepsilon < \infty$ et d'après (D.21),

$$ |y(t)| \le \max \{\varepsilon, \mu(\|u\|_{[t_0,t)})\} \le \max \{\bar{\eta}c + \beta(r, t - t_0), \mu(\|u\|_{[t_0,t)})\}. \qquad (D.23) $$
D'autre part, si $\varepsilon - \bar{\eta}c > \delta^{-1}(r)$, alors $\delta^{-1}(r) \le \beta(r,t - t_0)$ et, d'après (D.18),

$$|y(t)| \le \max \{\beta(r,t - t_0), \mu(\|u\|_{[t_0,t)}), c\}. \quad (D.24)$$

En combinant (D.23), (D.24), et en invoquant la Proposition A.1.2 :

$$ \begin{aligned} |y(t)| &\le \max \{(1 + \nu^{-1})\beta(r,t - t_0), \mu(\|u\|_{[t_0,t)}), (1 + \nu)\bar{\eta}c\} \\ &= \max \{\bar{\beta}(|x(t_0)|, t - t_0), \mu(\|u\|_{[t_0,t)}), \eta c\}, \end{aligned} $$

où $\bar{\beta} = (1 + \nu^{-1})\beta \in \mathcal{KL}$.

□

La proposition ci-dessous est une version locale du Théorème 1 dans [184]. C'est ce résultat qui nous permettra de clore les preuves de la section suivante.

**Proposition D.1.3** *Supposons qu'il existe $\beta \in \mathcal{KL}$, $\gamma^y, \gamma^u, \sigma^t, \sigma^x, \sigma^y$, $\sigma^u \in \mathcal{K}$, $c_x, c_y, m, M, \Delta \in \mathbb{R}_{>0}$ tels que, pour tout $x(t_0) \in \mathbb{R}^{n_x}$ et $u \in \mathcal{L}_{n_u}^\infty$ avec $\max \{|x(t_0)|, \|u\|_\infty\} < \Delta$, les solutions de (D.1)-(D.3) vérifient, pour tout $t \ge t_0 \ge 0$ :*

$$|y(t)| \le \max \{\beta(|x(t_0)|, t - t_0), \gamma^y(\|y\|_{[t_0,t)}), \gamma^u(\|u\|_{[t_0,t)}), c_y\} \quad (D.25)$$

$$|y(t)| \le M \quad (D.26)$$

$$|x(t)| \le \max \{\sigma^t(t - t_0), \sigma^x(|x(t_0)|), \sigma^y(\|y\|_{[t_0,t)}), \sigma^u(\|u\|_{[t_0,t)}), c_x\}, \quad (D.27)$$

et

$$\gamma^y(s) < s \quad \forall s \in [m, M], \quad (D.28)$$

avec

$$c = \max \{m, c_y\} < M, \quad (D.29)$$

soient

$$\tilde{\gamma}^u(s) = \max \{\beta(\sigma^y \circ \gamma^u(s), 0), \beta(\sigma^u(s), 0), \gamma^u(s)\} \quad \forall s \in \mathbb{R}_{\ge 0} \quad (D.30)$$

$$\bar{\Delta} \in (0, \Delta] \quad \text{avec} \quad \max \{\beta(\bar{\Delta}, 0), \tilde{\gamma}^u(\bar{\Delta})\} < M, \quad (D.31)$$

alors, pour tout $\eta \in (1, \infty)$, il existe $\tilde{\beta} \in \mathcal{KL}$ tel que pour tout $x(t_0) \in \mathbb{R}^{n_x}$ et $u \in \mathcal{L}_{n_u}^\infty$ avec $\max \{|x(t_0)|, \|u\|_\infty\} < \bar{\Delta}$ :

$$|y(t)| \le \max \{\tilde{\beta}(|x(t_0)|, t - t_0), \tilde{\gamma}^u(\|u\|_{[t_0,t)}), \eta c\} \quad \forall t \ge t_0 \ge 0. \quad (D.32)$$

**Preuve.** La preuve suit les mêmes étapes que celle du Théorème 1 dans [184]. La propriété de stabilité (D.32) est prouvée en invoquant la Proposition D.1.2. Nous allons donc démontrer que le système (D.1)-(D.3) est $(\tilde{\gamma}^u, c, \bar{\Delta})$-uniformément pratiquement stable entrée-sortie.

### Etape 1. Stabilité $(\tilde{\gamma}^u, c, \bar{\Delta})$-uniforme pratique.

Soient $\vartheta(s) = \beta^{-1}(s, 0)$ et $\delta(s) = \min \{\vartheta^{-1}(s), s\}$ pour $s \in \mathbb{R}_{\ge 0}$ (avec la convention consistante $\vartheta^{-1}(s) = \infty$ si $s \ge \sup_{\zeta \in [0, \infty)} \beta(\zeta, 0)$), nous savons que $\delta \in \mathcal{K}_\infty$. Soient $t_0 \in \mathbb{R}$,
$\varepsilon \in (0, \delta^{-1}(\Delta))$, $x(t_0) \in \mathbb{R}^{n_x}$ et $u \in \mathcal{L}_{\infty}^{n_u}$ avec $\max \{|x(t_0)|, \|u\|_{\infty}\} < \delta(\varepsilon)$. Premièrement, on remarque que, d'après (D.28) et le fait que $\gamma^y \in \mathcal{K}$,

$$ \left\{ \begin{array}{ll} \gamma^y(s) \le \gamma^y(m) < m & \text{si } s \in [0,m) \\ \gamma^y(s) < s & \text{si } s \in [m,M]. \end{array} \right. \qquad (D.33) $$

On en déduit, d'après (D.25), (D.26), (D.29) et (D.33), que :

$$ \begin{aligned} |y(t)| & \le \max \{\beta(|x(t_0)|,0), \gamma^y(\|y\|_{[t_0,t)}), \gamma^u(\|u\|_{[t_0,t)}), c_y\} \\ \|y\|_{[t_0,t)} & \le \max \{\beta(\delta(\varepsilon),0), \gamma^u(\|u\|_{[t_0,t)}), c\}, \end{aligned} \qquad (D.34) $$

en remarquant que,

$$ \begin{aligned} \left( \varepsilon \in [0, \sup_{\varsigma \in [0,\infty)} \beta(\varsigma,0)) \right) &\Rightarrow \left( \beta(\delta(\varepsilon),0) \le \beta(\vartheta^{-1}(\varepsilon),0) \le \varepsilon \right) \\ \left( \varepsilon \ge \sup_{\varsigma \in [0,\infty)} \beta(\varsigma,0) \right) &\Rightarrow \left( \beta(\delta(\varepsilon),0) \le \sup_{\varsigma \in [0,\infty)} \beta(\varsigma,0) \le \varepsilon \right), \end{aligned} $$

il s'en suit

$$ |y(t)| \le \|y\|_{[t_0,t)} \le \max \{\varepsilon, \gamma^u(\|u\|_{[t_0,t)}), c\}. $$

Ainsi, le système (D.1)-(D.3) est $(\tilde{\gamma}^u, c, \Delta)$-uniformément pratiquement stable et donc $(\tilde{\gamma}^u, c, \bar{\Delta})$-uniformément pratiquement stable (puisque $\bar{\Delta} \le \Delta$) d'après la Définition D.1.1.

### Étape 2. Attractivité $(\tilde{\gamma}^u, c, \bar{\Delta})$-uniforme pratique.

Soient $t_0 \in \mathbb{R}$, $r \in (0, \bar{\Delta})$, $\varepsilon \in (c, \infty)$, $x(t_0) \in \mathbb{R}^{n_x}$ et $u \in \mathcal{L}_{\infty}^{n_u}$ avec $\max \{|x(t_0)|, \|u\|_{\infty}\} < r$. Parallèlement à (D.34),

$$ \begin{aligned} |y(t)| & \le \max \{\beta(|x(t_0)|,0), \gamma^y(\|y\|_{[t_0,t)}), \gamma^u(\|u\|_{[t_0,t)}), c_y\} \\ \|y\|_{[t_0,t)} & \le \max \{\beta(r,0), \gamma^u(\|u\|_{[t_0,t)}), c\}. \end{aligned} \qquad (D.35) $$

Plusieurs suites sont maintenant définies : $(T_i^r)_{i \in \mathbb{Z}_{\ge 0}}$, $(\hat{T}_i^r)_{i \in \mathbb{Z}_{\ge 0}}$ et $(M_i^r)_{i \in \mathbb{Z}_{\ge 0}}$. Soient $T_0^r = 0$, $M_0^r = r$, $T_1^r > 0$ est choisi suffisamment grand tel que $\beta(M_0^r, T_1^r) \le \gamma^y(\beta(r,0))$. Soit $\hat{\sigma}(s) = \max\{\sigma^x(s), \sigma^y(\beta(s,0))\}$ pour $s \in \mathbb{R}_{\ge 0}$, $M_i^r$ est défini par, pour $i \in \mathbb{Z}_{>0}$ :

$$ M_i^r = \max \{\hat{\sigma}(r), \sigma^t(T_0^r + \dots + T_i^r), \sigma^y(c), c_x\}. \qquad (D.36) $$

D'autre part, $T_{i+1}^r > 0$ est choisi suffisamment grand tel que

$$ \beta(M_i^r, T_{i+1}^r) \le (\gamma^y)^{i+1}(\beta(r,0)), \qquad (D.37) $$

et $\hat{T}_i^r = T_0^r + \dots + T_i^r$, pour $i \in \mathbb{Z}_{\ge 0}$.
De par (D.27) et (D.35), on a, pour tout $t \in [t_0, \infty)$ :

$$
\begin{aligned}
|x(t)| & \le \max \left\{ \sigma^t(t - t_0), \sigma^x(r), \sigma^y(\beta(r,0)), \sigma^y(\gamma^u(\|u\|_{[t_0,t]})), \sigma^y(c), \right. \\
& \phantom{\le \max \left\{} \left. \sigma^u(\|u\|_{[t_0,t]}), c_x \right\} \\
& \le \max \left\{ \sigma^t(t - t_0), \hat{\sigma}(r), \sigma^y(\gamma^u(\|u\|_{[t_0,t]})), \sigma^u(\|u\|_{[t_0,t]}), \sigma^y(c), c_x \right\},
\end{aligned}
\quad (D.38)
$$

par définition des $M_i^r$ et $\hat{T}_i^r$, on déduit que :

$$
|x(t_0 + \hat{T}_i^r)| \le \max \left\{ M_i^r, \sigma^y(\gamma^u(\|u\|_{[t_0,t]})), \sigma^u(\|u\|_{[t_0,t]}) \right\}. \quad (D.39)
$$

Le fait ci-dessous nous permet de déduire la propriété d'attractivité désirée.

**Fait D.1.2** *Pour tout $i \in \mathbb{Z}_{\ge 0}$, les solutions de (D.1)-(D.3) satisfont :*

$$
|y(t)| \le \max \left\{ (\gamma^y)^i (\beta(r,0)), \tilde{\gamma}^u (\|u\|_{[t_0,t]}), c \right\} \quad \forall t \in [t_0 + \hat{T}_i^r, \infty). \quad (D.40)
$$

Preuve du fait. La preuve est réalisée par récurrence sur $i$. Premièrement, notons que pour $i=0$, (D.40) est assuré d'après (D.35). Supposons que (D.40) soit satisfait pour $i \in \mathbb{Z}_{\ge 0}$ i.e. pour $t \in [t_0 + \hat{T}_i^r, \infty)$ :

$$
|y(t)| \le \max \left\{ (\gamma^y)^i (\beta(r,0)), \tilde{\gamma}^u (\|u\|_{[t_0,t]}), c \right\}. \quad (D.41)
$$

Pour $t \in [t_0 + \hat{T}_{i+1}^r, \infty)$, d'après (D.25), (D.37) et (D.39),

$$
\begin{aligned}
|y(t)| & \le \max \left\{ \beta(|x(t_0 + \hat{T}_i^r)|, t - (t_0 + \hat{T}_i^r)), \gamma^y(\|y\|_{[t_0+\hat{T}_i^r, t]}), \gamma^u(\|u\|_{[t_0+\hat{T}_i^r, t]}), c_y \right\} \\
& \le \max \left\{ \beta(M_i^r, t - (t_0 + \hat{T}_i^r)), \tilde{\gamma}^u(\|u\|_{[t_0, t]}), \gamma^y(\|y\|_{[t_0+\hat{T}_i^r, t]}), c_y \right\} \\
& \le \max \left\{ (\gamma^y)^{i+1} (\beta(r,0)), \tilde{\gamma}^u(\|u\|_{[t_0, t]}), \gamma^y(\|y\|_{[t_0+\hat{T}_i^r, t]}) \right\},
\end{aligned}
\quad (D.42)
$$

Deux cas doivent être distingués.

$$
\text{Cas 1: } \gamma^y(\|y\|_{[t_0+\hat{T}_i^r, t]}) < \max \left\{ (\gamma^y)^{i+1}(\beta(r,0)), \tilde{\gamma}^u(\|u\|_{[t_0, t]}), c_y \right\}.
$$

Pour $t \in [t_0 + \hat{T}_{i+1}^r, \infty)$,

$$
|y(t)| \le \max \left\{ (\gamma^y)^{i+1} (\beta(r,0)), \tilde{\gamma}^u (\|u\|_{[t_0,t]}), c_y \right\}
$$

et (D.40) est satisfait puisque $c_y \le c$.

$$
\text{Cas 2: } \gamma^y(\|y\|_{[t_0+\hat{T}_i^r, t]}) \ge \max \left\{ (\gamma^y)^{i+1}(\beta(r,0)), \tilde{\gamma}^u(\|u\|_{[t_0,t]}), c_y \right\}.
$$

D'après (D.41) et (D.42), pour $t \in [t_0 + \hat{T}_{i+1}^r, \infty)$,

$$
\begin{aligned}
|y(t)| & \le \gamma^y(\|y\|_{[t_0+\hat{T}_i^r, t]}) \\
& \le \gamma^y \left( \max \left\{ (\gamma^y)^i (\beta(r,0)), \tilde{\gamma}^u(\|u\|_{[t_0,t]}), c \right\} \right) \\
& = \max \left\{ (\gamma^y)^{i+1} (\beta(r,0)), \gamma^y \left( \tilde{\gamma}^u(\|u\|_{[t_0,t]}) \right), \gamma^y(c) \right\}
\end{aligned}
\quad (D.43)
$$
puisque $\|u\|_{[t_0, \infty)} < \bar{\Delta}$, on a $\tilde{\gamma}^u(\|u\|_{[t_0, \infty)}) < M$, et en invoquant (D.29) et (D.33), on obtient :

$$|y(t)| \le \|y\|_{[t_0, t)} \le \max \left\{ (\gamma^y)^{i+1} (\beta(r,0)), \tilde{\gamma}^u (\|u\|_{[t_0, t)}), c \right\}, \quad (D.44)$$

par conséquent (D.40) est satisfait.

<figure><img src="image_34.png" /></figure>

On peut voir que $\limsup_{i \to \infty} \text{sup}(\gamma^y)^i(\beta(r,0)) \le m$, puisque $\beta(r,0) \le \beta(\bar{\Delta},0) < M$ et (D.28), (D.33) sont vérifiés. En conséquence, il existe toujours $i_0 \in \mathbb{Z}_{\ge 0}$ tel que $(\gamma^y)^{i_0}(\beta(r,0)) < \varepsilon$, puisque $\varepsilon > c \ge m$. Ainsi, soit $T(r, \varepsilon) = \hat{T}_{i_0}^r$, d'après le Fait D.1.2, on obtient pour tout $t \in [t_0 + T(\varepsilon, r), \infty)$ :

$$|y(t)| \le \max \left\{ (\gamma^y)^{i_0} (\beta(r,0)), \tilde{\gamma}^u (\|u\|_{[t_0, t)}), c \right\} \le \max \{\varepsilon, \tilde{\gamma}^u (\|u\|_{[t_0, t)}), c\}. \quad (D.45)$$

Le système (D.1)-(D.3) est $(\tilde{\gamma}^u, c, \bar{\Delta})$-uniformément pratiquement attractif d'après la Définition D.1.1.

Puisque le système (D.1)-(D.3) est $(\tilde{\gamma}^u, c, \bar{\Delta})$-uniformément pratiquement stable et attractif, la preuve se conclut en invoquant la Proposition D.1.2.

<figure><img src="image_35.png" /></figure>

## D.2 Théorèmes du petit gain pour des systèmes non-paramétrés

Soient les systèmes :

$$\left\{ \begin{array}{ll} \dot{x}_1 = f_1(t, x_1, x_2, u_1) & \forall t \in [t_{i-1}, t_i] \\ x_1(t_i^+) = g_1(t_i, x_1(t_i), x_2(t_i), u_1(t_i)) & \end{array} \right. \quad (D.46)$$

$$\left\{ \begin{array}{ll} \dot{x}_2 = f_2(t, x_2, x_1, u_2) & \forall t \in [t_{i-1}, t_i] \\ x_2(t_i^+) = g_2(t_i, x_2(t_i), x_1(t_i), u_2(t_i)), & \end{array} \right. \quad (D.47)$$

où $x_1 \in \mathbb{R}^{n_{x_1}}$, $x_2 \in \mathbb{R}^{n_{x_2}}$, $f_1 : \mathbb{R}_{\ge 0} \times \mathbb{R}^{n_{x_1}+n_{x_2}+n_{u_1}} \to \mathbb{R}^{n_{x_1}}$ et $f_2 : \mathbb{R}_{\ge 0} \times \mathbb{R}^{n_{x_2}+n_{x_1}+n_{u_2}} \to \mathbb{R}^{n_{x_2}}$ sont continues par morceaux en $t$ et continues en $x_1, x_2, u_1$ ou $u_2$, $g_1 : \mathbb{R}_{\ge 0} \times \mathbb{R}^{n_{x_1}+n_{x_2}+n_{u_1}} \to \mathbb{R}^{n_{x_1}}$ et $g : \mathbb{R}_{\ge 0} \times \mathbb{R}^{n_{x_2}+n_{x_1}+n_{u_2}} \to \mathbb{R}^{n_{x_2}}$ sont localement bornées telles que pour tout $(t, x_1, x_2, u_1, u_2) \in \mathbb{R}_{\ge 0} \times \mathbb{R}^{n_{x_1}+n_{x_2}+n_{u_1}+n_{u_2}}$ :

$$\begin{aligned} |g_1(t, x_1, x_2, u_1)| &\le |x_1| \\ |g_2(t, x_2, x_1, u_2)| &\le |x_2|. \end{aligned} \quad (D.48)$$

**Remarque D.2.1** Puisque notre analyse se base sur des bornes sur les trajectoires du système et non des fonctions de Lyapunov, on pourrait croire qu'il suffirait d'énoncer nos résultats pour des systèmes continus afin d'obtenir l'équivalent pour les systèmes hybrides à l'instar de [119]. Cependant, puisque la condition du petit gain ne sera vérifiée que sur un ensemble bornée, nos preuves requièrent des propriétés de continuité des variables d'état en fonction du temps. Nous verrons que les conditions (D.48) nous permettent tout de même de conclure. On remarque que les classes de systèmes étudiées dans la Partie II les vérifient.
Les instants de sauts $\{t_i\}_{i \in \mathbb{Z}_{>0}}$ vérifient, pour tout $i \in \mathbb{Z}_{>0}$ :

$$v \leq t_i - t_{i-1} \leq \tau, \qquad (D.49)$$

où $v \in \mathbb{R}_{>0}$ est une constante arbitrairement petite uniquement introduite afin d'éviter l'ap-
partition du paradoxe de Zénon, $\tau \in [v, \infty)$ est fixé et $t_0 \in \mathbb{R}_{\ge 0}$ est l'instant initial. On
définit les fonctions suivantes $\omega_1 \in C(\mathbb{R}^{n_{x_1}}, \mathbb{R}^{n_{\omega_1}})$, $\omega_2 = (\omega_2^a, \omega_2^b)$ où $\omega_2^a \in C(\mathbb{R}^{n_{x_2}}, \mathbb{R}^{n_{\omega_2^a}})$,
$\omega_2^b \in C(\mathbb{R}^{n_{x_2}}, \mathbb{R}^{n_{\omega_2^b}})$ avec $n_{\omega_1}, n_{\omega_2^a}, n_{\omega_2^b} \in \mathbb{Z}_{>0}$, telles que

$$|\omega_1(x_1)| \leq \rho_1(|x_1|) \qquad (D.50)$$

$$\underline{\rho}_2(|x_2|) \leq |\omega_2(x_2)| \leq \bar{\rho}_2(|x_2|), \qquad (D.51)$$

où $\rho_1, \underline{\rho}_2, \bar{\rho}_2 \in \mathcal{K}_\infty$.

Le premier théorème considère une interconnexion classique de deux systèmes qui vérifient des propriétés de type stabilité entrée-état pour lesquels la condition du petit gain est vérifiée sur un compact qui ne comprend pas l'origine. En comparaison avec [86, 197], nous proposons des estimés de la boule de conditions initiales et de l'erreur résiduelle obtenue en fonction des bornes de ce compact,

**Théorème D.2.1** *Considérons le système (D.46)-(D.47), supposons qu'il existe $\beta_1, \beta_2 \in \mathcal{KL}$, $\gamma_1^{\omega_2^a}, \gamma_1^u, \gamma_2^{\omega_1^1}, \gamma_2^u, \alpha_1, \eta_1^t, \eta_1^{\omega_1^1}, \eta_1^{\omega_2^a}, \eta_1^{\omega_2^b}, \eta_1^u, \alpha_2, \eta_2^{\omega_1^1}, \eta_2^{\omega_2^a}, \eta_2^u \in \mathcal{K}$ tels que pour tout $x_1(t_0) \in \mathbb{R}^{n_{x_1}}$, $(x_2, u_1) \in \mathcal{L}_{\infty}^{n_{x_2}+n_{u_1}}$, $t \ge t_0 \ge 0$, les solutions de (D.46) vérifient :*

$$|\omega_1(x_1(t))| \leq \max \{\beta_1(|\omega_1(x_1(t_0))|, t - t_0), \gamma_1^{\omega_2^a}(\|\omega_2^a(x_2)\|_{[t_0,t)}), \gamma_1^u(\|u_1\|_{[t_0,t)})\} \quad (D.52)$$

$$|x_1(t)| \leq \max \{ \alpha_1(|x_1(t_0)|), \eta_1^t(t - t_0), \eta_1^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t)}), \eta_1^{\omega_2^a}(\|\omega_2^a(x_2)\|_{[t_0,t)}), \\ \eta_1^{\omega_2^b}(\|\omega_2^b(x_2)\|_{[t_0,t)}), \eta_1^u(\|u_1\|_{[t_0,t)}) \}, \quad (D.53)$$

pour tout $x_2(t_0) \in \mathbb{R}^{n_{x_2}}$, $(x_1, u_2) \in \mathcal{L}_{\infty}^{n_{x_1}+n_{u_2}}$, $t \ge t_0 \ge 0$, les solutions de (D.47) vérifient :

$$|\omega_2^a(x_2(t))| \leq \max \{\beta_2(|\omega_2^a(x_2(t_0))|, t - t_0), \gamma_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t)}), \gamma_2^u(\|u_2\|_{[t_0,t)})\} \quad (D.54)$$

$$|\omega_2^b(x_2(t))| \leq \max \{ \alpha_2(|\omega_2^b(x_2(t_0))|), \eta_2^t(t - t_0), \eta_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t)}), \\ \eta_2^{\omega_2^a}(\|\omega_2^a(x_2)\|_{[t_0,t)}), \eta_2^u(\|u_2\|_{[t_0,t)}) \}, \quad (D.55)$$

et qu'il existe $0 < m < M$ tels que

$$\max \{\gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1^1}(s), \gamma_2^{\omega_1^1} \circ \gamma_1^{\omega_2^a}(s)\} < s \quad \text{pour } s \in [m, M], \qquad (D.56)$$

sous la condition suivante

$$\tilde{\sigma}^m(m) < M, \qquad (D.57)$$

où

$$\tilde{\sigma}^m(s) = \max \{ \beta_1(s,0), \gamma_1^{\omega_2^a}(\beta_2(s,0)), \beta_2(s,0), \gamma_2^{\omega_1^1}(\beta_1(s,0)) \}. $$
On définit les fonctions de classe $\mathcal{K} \nu^x, \nu^u$, pour $s \in \mathbb{R}_{\ge 0}$ :

$$ \nu^x(s) = \max \{ \beta_1(\rho_1(s),0), \gamma_1^{\omega_2^a}(\beta_2(\bar{\rho}_2(s),0)), \rho_1(s), \beta_2(\bar{\rho}_2(s),0), \gamma_2^{\omega_1^a}(\beta_1(\rho_1(s),0)), \\ \qquad \tilde{\beta}(s,0), s \} $$

$$ \nu^u(s) = \max \{ \gamma_1^u(s), \gamma_1^{\omega_2^a} \circ \gamma_2^u(s), \gamma_2^u(s), \gamma_2^{\omega_1^a} \circ \gamma_1^u(s), \tilde{\gamma}^u(s) \}, $$

où $\tilde{\beta} \in \mathcal{KL}$, $\tilde{\gamma}^u \in \mathcal{K}$ sont définies respectivement par (D.74) et (D.78), et soit $\Delta \in \mathbb{R}_{>0}$ tel que $\max \{\nu^x(\Delta), \nu^u(\Delta)\} < M$.

Alors, le système (D.46)-(D.47) est uniformément positivement complet pour tout $(x_1(t_0), x_2(t_0)) \in \mathbb{R}^{n_{x_1}+n_{x_2}}$ et $(u_1,u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ avec $\max \{|x_1(t_0)|,|\omega_2^a(x_2(t_0))|, \|u_1\|_{\infty}, \|u_2\|_{\infty}\} < \Delta$. De plus, pour tout $\eta > 1$, il existe $\beta \in \mathcal{KL}$, $\sigma \in \mathcal{K}$ telles que pour tout $(x_1(t_0),x_2(t_0)) \in \mathbb{R}^{n_{x_1}+n_{x_2}}$ et $(u_1,u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ avec $\max \{|x_1(t_0)|,|\omega_2^a(x_2(t_0))|, \|u_1\|_{\infty}, \|u_2\|_{\infty}\} < \Delta$, les solutions de (D.46)-(D.47) satisfont, pour tout $t \ge t_0 \ge 0$,

$$ \max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max \{ \beta(\max \{|\omega_1(x_1(t_0))|, |\omega_2^a(x_2(t_0))|\}, t - t_0), \\ \sigma(\max \{ \|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)} \}), \delta(m) \}, \quad (D.58) $$

où $\delta(s) = \max \{\eta\tilde{\sigma}^m(s), \eta s\}$, pour $s \in \mathbb{R}_{\ge 0}$.

**Preuve.** Il est tout d'abord prouvé que le système est uniformément positivement complet lorsque les conditions initiales et les entrées $(u_1,u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ sont convenablement majorées. La propriété de stabilité désirée est ensuite obtenue en invoquant la Proposition D.1.3.

### Etape 1. Compléture positive et majoration de $\omega_1(x_1)$ et $\omega_2^a(x_2)$.

Soient $t_0 \in \mathbb{R}_{\ge 0}$, $(x_1(t_0),x_2(t_0)) \in \mathbb{R}^{n_{x_1}+n_{x_2}}$ et $(u_1,u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ tels que

$$ \max \{|x_1(t_0)|, |\omega_2^a(x_2(t_0))|, \|u_1\|_{\infty}, \|u_2\|_{\infty}\} < \Delta, \quad (D.59) $$

ce qui implique, d'après la définition de $\Delta$,

$$ \nu^x(\max \{|x_1(t_0)|, |\omega_2^a(x_2(t_0))|\}) < M \quad (D.60) $$

$$ \nu^u(\max \{ \|u_1\|_{[t_0,\infty)}, \|u_2\|_{[t_0,\infty)} \}) < M. \quad (D.61) $$

Soit $[t_0,t_{\max})$ l'intervalle maximal d'existence de (D.46)-(D.47), où $t_{\max} \in (t_0,\infty]$. De par (D.52) et (D.54), on a, pour tout $t \in [t_0,t_{\max})$ :

$$ |\omega_1(x_1(t))| \le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, t - t_0), \gamma_1^{\omega_2^a}(\beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \\ \qquad \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1^a}(\|\omega_1(x_1)\|_{[t_0,t)})), \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t)})), \\ \qquad \left. \gamma_1^u(\|u_1\|_{[t_0,t)}) \right\} \quad (D.62) $$

$$ \|\omega_1(x_1)\|_{[t_0,t)} \le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, 0), \gamma_1^{\omega_2^a}(\beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \\ \qquad \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1^a}(\|\omega_1(x_1)\|_{[t_0,t)})), \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t)})), \\ \qquad \left. \gamma_1^u(\|u_1\|_{[t_0,t)}) \right\}. \quad (D.63) $$
Nous allons montrer par l'absurde que $|\omega_1(x_1(t))| < M$ pour tout $t \in [t_0, t_{\max})$. Dans un premier temps, on remarque que $|\omega_1(x_1(t_0))| < M$, d'après (D.50) et (D.60). On suppose qu'il existe $t_1 \in (t_0, t_{\max})$ tel que $|\omega_1(x_1(t_1))| = M$. Soit $\bar{t} = \inf\{t \in (t_0, t_{\max}) : |\omega_1(x_1(t))| = M\}$. De par (D.63), on a :

$$
\begin{align*}
\|\omega_1(x_1)\|_{[t_0, \bar{t})} = |\omega_1(x_1(\bar{t}))| &\le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, 0), \gamma_1^{\omega_2^a} (\beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \\
&\phantom{\le} \gamma_1^{\omega_2^a} (\gamma_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0, \bar{t})}), \gamma_1^{\omega_2^a} (\gamma_2^u(\|u_2\|_{[t_0, \bar{t})})), \\
&\phantom{\le} \gamma_1^u (\|u_1\|_{[t_0, \bar{t})}) \right\} \\
&= \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, 0), \gamma_1^{\omega_2^a} (\beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \\
&\phantom{= \max \left\{ \right.} \gamma_1^{\omega_2^a} (\gamma_2^{\omega_1}(M)), \gamma_1^{\omega_2^a} (\gamma_2^u(\|u_2\|_{[t_0, \bar{t})})), \\
&\phantom{= \max \left\{ \right.} \gamma_1^u (\|u_1\|_{[t_0, \bar{t})}) \right\}. \tag{D.64}
\end{align*}
$$

Notons que, d'après (D.60) et (D.61) :

$$
\max \left\{ \beta_1(|\omega_1(x_1(t_0))|,0), \gamma_1^{\omega_2^a} (\beta_2(|\omega_2^a(x_2(t_0))|,0)), \gamma_1^{\omega_2^a} (\gamma_2^u (\|u_2\|_{[t_0,\bar{t})})), \gamma_1^u (\|u_1\|_{[t_0,\bar{t})}) \right\} < M, \quad (D.65)
$$

de plus, en vu de (D.56) :

$$
\gamma_1^{\omega_2^a}(\gamma_2^{\omega_1}(M)) < M, \tag{D.66}
$$

par conséquent, d'après (D.64), (D.65), (D.66),

$$
|\omega_1(x_1(\bar{t}))| < M
$$

ce qui contredit le fait que $|\omega_1(x_1(\bar{t}))| = M$, ainsi puisque $\omega_1(x_1)$ est continue entre deux instants de saut et d'après (D.48), pour tout $t \in [t_0, t_{\max})$ :

$$
|\omega_1(x_1(t))| < M. \tag{D.67}
$$

Par analogie, on peut montrer que pour tout $t \in [t_0, t_{\max})$ :

$$
|\omega_2^a(x_2(t))| < M. \tag{D.68}
$$

En conséquence, en invoquant (D.51), (D.55), (D.53), (D.67) et (D.68), on obtient par que $t_{\max} = \infty$ par l'absurde et on peut montrer que le système (D.46)-(D.47) est uniformément positivement complet d'après la Définition A.2.3.

Etape 2. Propriété (D.58).

Nous allons démontrer que les conditions de la Proposition D.1.3 sont vérifiées. Premièrement, notons que d'après (D.56) et puisque $\gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1} \in \mathcal{K}$ :

$$
\left\{
\begin{array}{ll}
\gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1}(s) \le \gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1}(m) < m & \text{si } s \in [0, m) \\
\gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1}(s) < s & \text{si } s \in [m, M].
\end{array}
\right.
\tag{D.69}
$$
Ainsi, pour $t \in [t_0, \infty)$, on a d'après (D.62), (D.67) et (D.69) :

$$
\begin{align}
|\omega_1(x_1(t))| &\le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, t-t_0), \gamma_1^{\omega_2^a}(\beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \nonumber \\
&\phantom{{}\le \max \left\{ \right.} \left. \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1}(\|\omega_1(x_1(t_0))\|_{[t_0,t]})), \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]})), \gamma_1^u(\|u_1\|_{[t_0,t]}) \right\} \nonumber \\
\|\omega_1(x_1)\|_{[t_0,t)} &\le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, 0), \gamma_1^{\omega_2^a}(\beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \nonumber \\
&\phantom{{}\le \max \left\{ \right.} \left. \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]})), \gamma_1^u(\|u_1\|_{[t_0,t]}), m \right\}, \tag{D.70}
\end{align}
$$

par symétrie,

$$
\|\omega_2^a(x_2)\|_{[t_0,t)} \le \max \left\{ \beta_2(|\omega_2^a(x_2(t_0))|,0), \gamma_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0))|,0)), \right. \\
\qquad \left. \gamma_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]})), \gamma_2^u(\|u_2\|_{[t_0,t]}), m \right\}. \tag{D.71}
$$

D'autre part, d'après (D.54) pour tout $t \in [t_0, t_{\max})$ :

$$
|\omega_2^a(x_2(t))| \le \max \left\{ \beta_2(|\omega_2^a(x_2(\frac{t}{2})))|, \frac{t}{2} \right\}, \gamma_2^{\omega_1}(\|\omega_1(x_1(t_0))\|_{[\frac{t}{2},t)}), \gamma_2^u(\|u_2\|_{[\frac{t}{2},t)}) \right\}, \quad (D.72)
$$

par conséquent d'après (D.52), (D.70), (D.71) et (D.72) :

$$
\begin{align*}
|\omega_1(x_1(t))| &\le \max \left\{ \beta_1(|\omega_1(x_1(\frac{t}{2}))|, \frac{t}{2}), \gamma_1^{\omega_2^a}(\|\omega_2^a(x_2)\|_{[\frac{t}{2},t)}), \gamma_1^u(\|u_1\|_{[\frac{t}{2},t)}) \right\} \\
&\le \max \left\{ \beta_1(\beta_1(|\omega_1(x_1(t_0))|, 0), \frac{t}{2}), \beta_1(\gamma_1^{\omega_2^a}(\beta_2(|\omega_2^a(x_2(t_0))|, 0)), \frac{t}{2}), \right. \\
&\qquad \beta_1(\gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]})), \frac{t}{2}), \beta_1(\gamma_1^u(\|u_1\|_{[t_0,t]})), \frac{t}{2}), \\
&\qquad \beta_1(m, \frac{t}{2}), \gamma_1^{\omega_2^a}(\beta_2(\beta_2(|\omega_2^a(x_2(t_0))|, 0), \frac{t}{2})), \\
&\qquad \gamma_1^{\omega_2^a}(\beta_2(\gamma_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0))|, 0)), \frac{t}{2})), \\
&\qquad \gamma_1^{\omega_2^a}(\beta_2(\gamma_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]})), \frac{t}{2})), \gamma_1^{\omega_2^a}(\beta_2(\gamma_2^u(\|u_2\|_{[t_0,t]})), \frac{t}{2})), \\
&\qquad \gamma_1^{\omega_2^a}(\beta_2(m, \frac{t}{2})), \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1}(\|\omega_1(x_1)\|_{[\frac{t}{2},t]})), \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[\frac{t}{2},t]})), \\
&\qquad \gamma_1^u(\|u_1\|_{[\frac{t}{2},t]}).
\end{align*}
$$

Un résultat analogue est vérifié par $\omega_2^a(x_2)$, ainsi,

$$
\max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max \left\{
\begin{array}{@{}l@{}}
  \tilde{\beta}(\max \{|\omega_1(x_1(t_0))|, |\omega_2^a(x_2(t_0))|\}, t - t_0), \\
  \tilde{\sigma}^u(\max \{\|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)}\}), \tilde{\sigma}^m(m), \\
  \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1}(\|\omega_1(x_1)\|_{[0,t]})), \\
  \gamma_2^{\omega_1}(\gamma_1^{\omega_2^a}(\|\omega_1(x_1)\|_{[0,t]}))
\end{array}
\right\}, \quad (D.73)
$$
où

$$
\tilde{\beta}(s,t) = \max \left\{ \begin{array}{l} \beta_1(\beta_1(s,0), \frac{t}{2}), \beta_1(\gamma_1^{\omega_2^a}(\beta_2(s,0)), \frac{t}{2}), \gamma_1^{\omega_2^a}(\beta_2(\beta_2(s,0), \frac{t}{2})), \\ \gamma_1^{\omega_2^a}(\beta_2(\gamma_2^{\omega_1^1}(\beta_1(s,0)), \frac{t}{2})) \\ \beta_2(\beta_2(s,0), \frac{t}{2}), \beta_2(\gamma_2^{\omega_1^1}(\beta_1(s,0)), \frac{t}{2}), \gamma_2^{\omega_1^1}(\beta_1(\beta_1(s,0), \frac{t}{2})), \\ \gamma_2^{\omega_1^1}(\beta_1(\gamma_1^{\omega_2^a}(\beta_2(s,0)), \frac{t}{2})) \end{array} \right\}, \qquad (D.74)
$$

$$
\tilde{\sigma}^u(s) = \max \left\{ \begin{array}{l} \beta_1(\gamma_1^{\omega_2^a}(\gamma_2^u(s)),0), \beta_1(\gamma_1^u(s),0), \gamma_1^{\omega_2^a}(\beta_2(\gamma_2^{\omega_1^1}(\gamma_1^u(s)),0)), \\ \gamma_1^{\omega_2^a}(\beta_2(\gamma_2^u(s),0)), \gamma_1^{\omega_2^a}(\gamma_2^u(s)), \gamma_1^u(s) \\ \beta_2(\gamma_2^{\omega_1^1}(\gamma_1^u(s)),0), \beta_2(\gamma_2^u(s),0), \gamma_2^{\omega_1^1}(\beta_1(\gamma_1^{\omega_2^a}(\gamma_2^u(s)),0)), \\ \gamma_2^{\omega_1^1}(\beta_1(\gamma_1^u(s),0)), \gamma_2^{\omega_1^1}(\gamma_1^u(s)), \gamma_2^u(s) \end{array} \right\}, \qquad (D.75)
$$

$$
\tilde{\sigma}^m(s) = \max \{ \beta_1(s,0), \gamma_1^{\omega_2^a}(\beta_2(s,0)), \beta_2(s,0), \gamma_2^{\omega_1^1}(\beta_1(s,0)) \}. \quad (D.76)
$$

Soient, pour $s \in \mathbb{R}_{\ge 0}$,

$$
\sigma^y(s) = \max \{ \gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1^1}(s), \gamma_2^{\omega_1^1} \circ \gamma_1^{\omega_2^a}(s) \} \quad (D.77)
$$

$$
\tilde{\gamma}^u(s) = \max \{ \tilde{\beta}(\sigma^y \circ \tilde{\sigma}^u(s), 0), \tilde{\beta}(\tilde{\sigma}^u(s), 0), \tilde{\sigma}^u(s) \}. \quad (D.78)
$$

On constate que les conditions de la Proposition D.1.3 sont satisfaites. En effet, on identifie $x = y = \max \{|\omega_1(x_1)|,|\omega_2^a(x_2)|\}$, pour $(s,t) \in \mathbb{R}_{\ge 0}^2$, $\beta(s,t) = \tilde{\beta}(s,t)$, $\gamma^y(s) = \max \{\gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1^1}(s), \gamma_2^{\omega_1^1}(s)\}$, $\gamma^u(s) = \tilde{\sigma}^u(s)$, $c_y = \tilde{\sigma}^m(m)$, $\sigma^t(s) = 0$, $\sigma^x(s) = \tilde{\beta}(s,0)$, $\sigma^y$ est défini dans (D.77), $\sigma^u(s) = \tilde{\sigma}^u(s)$, $c_x = \tilde{\sigma}^m(m)$, $c = \max \{\tilde{\sigma}^m(m), m\}$, $\tilde{\gamma}^u$ est donné dans (D.78). L'inégalité (D.25) est satisfaite d'après (D.73), (D.26) d'après (D.67) et (D.68), (D.27) d'après (D.73), (D.28) d'après (D.56), (D.29) d'après (D.57), (D.31) avec $\bar{\Delta} = \Delta$ puisque $\max\{\nu^x(\Delta), \nu^u(\Delta)\} < M$. Par conséquent, puisque (D.59) est vérifiée, en invoquant la Proposition D.1.3, pour $\eta > 1$, il existe $\beta \in \mathcal{KL}$ telle que, pour tout $t \in [t_0, \infty)$ :

$$
\max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max \left\{
\begin{aligned}
& \beta(\max \{|\omega_1(x_1(t_0))|, |\omega_2^a(x_2(t_0))|\}, t - t_0), \\
& \tilde{\gamma}^u(\max \{\|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)}\}), \\
& \eta \tilde{\sigma}^m(m), \eta m
\end{aligned}
\right\}. \quad (D.79)
$$

La preuve se conclut en notant $\delta(s) = \max \{\eta \tilde{\sigma}^m(s), \eta s\}$ pour $s \in \mathbb{R}_{\ge 0}$.

<figure><img src="image_36.png" /></figure>

**Remarque D.2.2** La condition (D.57) signifie que les bornes du compact sur lequel la condi-
tion du petit gain (D.56) est satisfaite ne peuvent être trop proches. On remarque les auteurs
de [191] ont oublié de le préciser dans le Lemme 2.
Le théorème suivant est atypique puisqu'il s'agit d'une triple interconnexion pour laquelle un des sous-systèmes ne vérifient pas de propriété de stabilité asymptotique mais de type entrée-bornée-état-borné. Dans [40], des théorèmes du petit gain sont proposés pour des interconnexions multiples de systèmes. Il y est montré que la stabilité entrée-état, la propriété de gain asymptotique et la stabilité entrée-bornée-état-borné sont conservées une fois par le système global interconnecté sous de nouvelles conditions du petit gain. Contrairement à [40], les systèmes étudiés ici ont chacun des propriétés de stabilité différentes, de plus leur interconnexion est assurée d'être pratiquement stable et non pas asymptotiquement stable.

**Théorème D.2.2** *Considérons le système (D.46)-(D.47), supposons qu'il existe $\beta_1, \beta_2 \in \mathcal{KL}$, $\gamma_1^{\omega_1}, \gamma_1^{\omega_2}, \gamma_1^u, \gamma_2^{\omega_1}, \gamma_2^u, \alpha_1, \eta_1^{\omega_1}, \eta_1^{\omega_2}, \eta_1^{\omega_1}, \eta_2^{\omega_2}, \eta_2^{\omega_1}, \eta_2^{\omega_2} \in \mathcal{K}$ tels que pour tout $x_1(t_0) \in \mathbb{R}^{n_{x_1}}$, $(x_1, x_2) \in \mathcal{L}_{\infty}^{n_{x_1}+n_{x_2}}$, $t \ge t_0 \ge 0$, les solutions de (D.46) vérifient :*

$$ |\omega_1(x_1(t))| \le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, t-t_0), \gamma_1^{\omega_2}(\|\omega_2^a(x_2)\|_{[t_0,t)}), \gamma_1^{\omega_2}(\|\omega_2^b(x_2)\|_{[t_0,t)}), \right. \\ \left. \gamma_1^u(\|u_1\|_{[t_0,t)}) \right\} \quad (D.80) $$

$$ |x_1(t)| \le \max \left\{ \alpha_1(|x_1(t_0)|), \eta_1^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,t)}), \eta_1^{\omega_2}(\|\omega_2^a(x_2)\|_{[t_0,t)}), \right. \\ \left. \eta_1^{\omega_2}(\|\omega_2^b(x_2)\|_{[t_0,t)}), \eta_1^u(\|u_1\|_{[t_0,t)}) \right\}, \quad (D.81) $$

*pour tout $x_2(t_0) \in \mathbb{R}^{n_{x_2}}$, $(x_1, u_2) \in \mathcal{L}_{\infty}^{n_{x_1}+n_{x_2}}$, et $t \ge t_0 \ge 0$, les solutions de (D.47) vérifient :*

$$ |\omega_2^a(x_2(t))| \le \max \left\{ \beta_2(|\omega_2(x_2(t_0))|, t-t_0), \gamma_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,t)}), \gamma_2^u(\|u_2\|_{[t_0,t)}) \right\} \quad (D.82) $$

$$ |\omega_2^b(x_2(t))| \le \max \left\{ \alpha_2(|\omega_2^b(x_2(t_0))|), \eta_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,t)}), \eta_2^{\omega_2}(\|\omega_2^a(x_2)\|_{[t_0,t)}), \right. \\ \left. \eta_2^u(\|u_2\|_{[t_0,t)}) \right\}, \quad (D.83) $$

*et qu'il existe $0 < m < M$ tels que*

$$ \max \left\{ \gamma_1^{\omega_2} \circ \gamma_2^{\omega_1}(s), \gamma_1^{\omega_2} \circ \eta_2^{\omega_2} \circ \gamma_2^{\omega_1}(s), \gamma_1^{\omega_2} \circ \eta_2^{\omega_1}(s), \gamma_2^{\omega_1} \circ \gamma_1^{\omega_2}(s), \right. \\ \left. \gamma_2^{\omega_1} \circ \gamma_1^{\omega_2} \circ \eta_2^{\omega_2}(s), \gamma_2^{\omega_1} \circ \gamma_1^{\omega_2} \circ \eta_2^{\omega_1}(s) \right\} < s \quad \text{pour } s \in [m,M], $$

*avec*

$$ \tilde{\sigma}^m(m) < M, \quad (D.85) $$

où $\tilde{\sigma}^m$ est défini par (D.118). On définit les fonctions de classe $\mathcal{K} \nu^x, \nu^u$, pour $s \in \mathbb{R}_{\ge 0}$ :

$$ \nu^x(s) = \max \left\{ \beta_1(\rho_1(s),0), \gamma_1^{\omega_2}(\beta_2(\bar{\rho}_2(s),0)), \gamma_1^{\omega_2}(\alpha_2(\bar{\rho}_2(s))), \gamma_1^{\omega_2}(\eta_2^{\omega_2}(\beta_2(\bar{\rho}_2(s),0))), \right. \\ \left. \beta_2(\bar{\rho}_2(s),0), \gamma_2^{\omega_1}(\beta_1(\rho_1(s),0)), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\bar{\rho}_2(s)))), \rho_1(s), \bar{\rho}_2(s), \tilde{\beta}(s,0), \right. \\ \left. \tilde{\sigma}^x(s) \right\} $$

$$ \nu^u(s) = \max \left\{ \gamma_1^{\omega_2}(\gamma_2^u(s)), \gamma_1^u(s), \gamma_1^{\omega_2}(\eta_2^{\omega_2}(\gamma_2^u(s))), \gamma_1^{\omega_2}(\eta_2^u(s)), \gamma_2^{\omega_1}(\gamma_1^u(s)), \gamma_2^u(s), \right. \\ \left. \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))), \tilde{\gamma}^u(s) \right\}, $$

où $\tilde{\beta} \in \mathcal{KL}$, $\tilde{\sigma}^x, \tilde{\gamma}^u \in \mathcal{K}$ sont respectivement définies dans (D.116), (D.119) et (D.122).
Soit $\Delta \in \mathbb{R}_{>0}$ tel que $\max\{\nu^x(\Delta), \nu^u(\Delta)\} < M$, alors il existe $\mu \in \mathbb{R}_{>0}$ tel que, pour tout $(x_1(t_0), x_2(t_0)) \in \mathbb{R}^{n_{x_1}+n_{x_2}}$ et $(u_1, u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ avec $\max\{|x_1(t_0)|, |x_2(t_0)|, \|u_1\|_{\infty}, \|u_2\|_{\infty}\}$
< Δ, les solutions de (D.46)-(D.47) vérifient :

$$
|(x_1(t), x_2(t))| \le \mu \quad \forall t \ge t_0 \ge 0. \tag{D.86}
$$

De plus, pour tout η > 1, il existe β ∈ K L, σ ∈ K, (x₁(t₀), x₂(t₀)) ∈ ℝ^{n_{x₁} + n_{x₂}} et (u₁, u₂) ∈ L^{n_{u₁} + n_{u₂}} ∞ avec max {|x₁(t₀)|, |x₂(t₀)|, ||u₁||_∞, ||u₂||_∞} < Δ, les solutions de (D.46)-(D.47) satisfont, pour tout t ≥ t₀ ≥ 0,

$$
\begin{equation}
\begin{split}
\max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max & \left\{ \beta(\max \{|\omega_1(x_1(t_0))|, |\omega_2(x_2(t_0))|\}, t-t_0), \right. \\
& \qquad \left. \sigma(\max \{\|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)}\}), \right. \\
& \qquad \left. \delta^m(m), \delta^x(\Delta) \right\},
\end{split}
\tag{D.87}
\end{equation}
$$

où, pour s ∈ ℝ ≥ 0, δ $^{m}$( s ) = max { η ˜ σ $^{m}$( s ) , ηs } et δ $^{x}$( s ) = η ˜ σ $^{x}$( s ).

**Preuve.** Il est tout d’abord démontré que le système est uniformément positivement complet lorsque les conditions initiales et les entrées sont convenablement bornées. La propriété de stabilité désirée est ensuite prouvée à l’aide de la Proposition D.1.3.

Etape 1. Complétude positive, majoration de ω₁(x₁) et ω²ₐ(x₂) et bornitude des états.

Soient $t_0 \in \mathbb{R}_{\ge 0}$, $\Delta \in \mathbb{R}_{> 0}$ avec $\max \{\nu^x(\Delta), \nu^u(\Delta)\} < M$, $(x_1(t_0), x_2(t_0)) \in \mathbb{R}^{n_{x_1}+n_{x_2}}$ et $(u_1, u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ tels que

$$
\max \{|x_1(t_0)|, |x_2(t_0)|, \|u_1\|_{[t_0, \infty)}, \|u_2\|_{[t_0, \infty)} \} < \Delta. \quad (D.88)
$$

Soit $[t_0, t_{\max})$ l'intervalle maximal d'existence du système (D.46)-(D.47), où $t_{\max} \in (t_0, \infty]$.
D'après (D.50), (D.51), (D.80), (D.82) et (D.83), pour tout $t \in [t_0, t_{\max})$ :

$$
\begin{equation}
\begin{aligned}
|\omega_1(x_1(t))| &\le \max \left\{
  \begin{aligned}[t]
  \beta_1(|\omega_1(x_1(t_0))|, t-t_0), &\gamma_1^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0))|, 0)), \\
  \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,t)})), &\gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t)})), \\
  \gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|)), &\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\beta_2(|\omega_2(x_2(t_0))|, 0))), \\
  \gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\gamma_2^u(\|\omega_1(x_1)\|_{[t_0,t)}))), &\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t)}))), \\
  \gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\|\omega_1(x_1)\|_{[t_0,t)})), &\gamma_1^{\omega_2^b}(\eta_2^u(\|u_2\|_{[t_0,t)}))
  \end{aligned}
\right\}
\end{aligned}
\tag{D.89}
\end{equation}
$$

$$
\begin{align}
\|\omega_1(x_1)\|_{[t_0,t)} &\le \max \left\{
  \begin{aligned}[t]
  \beta_1(\rho_1(x_1(t_0)),0), & \gamma_1^{\omega_2^a}(\beta_2(\bar{\rho}_2(x_2(t_0)),0)), \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t)})), \\
  \gamma_1^{\omega_2^u}(\|u_1\|_{[t_0,t)}), & \gamma_1^{\omega_2^b}(\alpha_2(\bar{\rho}_2(x_2(t_0)))), \\
  \gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\beta_2(\bar{\rho}_2(x_2(t_0)),0))), & \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t)}))), \\
  \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1^u}(\|\omega_1(x_1)\|_{[t_0,t)})), & \gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\gamma_2^{\omega_1^u}(\|\omega_1(x_1)\|_{[t_0,t)}))), \\
  \gamma_1^{\omega_2^b}(\eta_2^{\omega_1^u}(\|\omega_1(x_1)\|_{[t_0,t)}))
  \end{aligned}
\right\}. \tag{D.90}
\end{align}
$$

Nous allons montrer par l'absurde que |ω₁(x₁(t))| < M pour tout t ∈ [t₀, tmax). Premièrement,
on note que |ω₁(x₁(t₀))| < M, d'après (D.50) et (D.88). On suppose qu'il existe t₁ ∈ (t₀, tmax)
tel que $|\omega_1(x_1(t_1))| = M$. Soit $\bar{t} = \inf\{t \in (t_0, t_{\max}) : |\omega_1(x_1(t))| = M\}$. D'après (D.90), on a :

$$
\begin{align*}
\|\omega_1(x_1)\|_{[t_0, \bar{t})} = |\omega_1(x_1(\bar{t}))| &\le \max \left\{ \beta_1(\rho_1(x_1(t_0)),0), \gamma_1^{\omega_2^a}(\beta_2(\bar{\rho}_2(x_2(t_0)),0)), \right. \\
&\qquad \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,\bar{t})})), \gamma_1^u(\|u_1\|_{[t_0,\bar{t})}), \\
&\qquad \gamma_1^{\omega_2^b}(\alpha_2(\bar{\rho}_2(x_2(t_0)))), \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(\bar{\rho}_2(x_2(t_0)),0))), \\
&\qquad \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,\bar{t})}))), \gamma_1^{\omega_2^b}(\eta_2^u(\|u_2\|_{[t_0,\bar{t})})), \\
&\qquad \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,\bar{t})})), \\
&\qquad \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,\bar{t})}))), \\
&\qquad \gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,\bar{t})})) \biggr\} \\
&= \max \left\{ \beta_1(\rho_1(x_1(t_0)),0), \gamma_1^{\omega_2^a}(\beta_2(\bar{\rho}_2(x_2(t_0)),0)), \right. \\
&\qquad \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,\bar{t})})), \gamma_1^u(\|u_1\|_{[t_0,\bar{t})}), \\
&\qquad \gamma_1^{\omega_2^b}(\alpha_2(\bar{\rho}_2(x_2(t_0)))), \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(\bar{\rho}_2(x_2(t_0)),0))), \\
&\qquad \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,\bar{t})}))), \gamma_1^{\omega_2^b}(\eta_2^u(\|u_2\|_{[t_0,\bar{t})})), \\
&\qquad \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1}(M)), \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(M))), \\
&\qquad \gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(M)) \biggr\}. \tag{D.91}
\end{align*}
$$

On remarque que, d'après (D.88) :

$$
\max \left\{ \beta_1(\rho_1(x_1(t_0)),0), \gamma_1^{\omega_2^a}(\beta_2(\bar{\rho}_2(x_2(t_0)),0)), \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,\bar{t})})), \gamma_1^u(\|u_1\|_{[t_0,\bar{t})}), \right. \\
\gamma_1^{\omega_2^b}(\alpha_2(\bar{\rho}_2(x_2(t_0)))), \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(\bar{\rho}_2(x_2(t_0)),0))), \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,\bar{t})}))), \tag{D.92} \\
\left. \gamma_1^{\omega_2^b}(\eta_2^u(\|u_2\|_{[t_0,\bar{t})})) \right\} < M,
$$

en conséquence, d'après (D.84), (D.91), (D.92),

$$
|\omega_1(x_1(\bar{t}))| < M,
$$

ce qui contredit le fait que $|\omega_1(x_1(\bar{t}))| = M$, par conséquent, puisque $\omega_1(x_1)$ est continue entre deux instants de saut et d'après (D.48), pour tout $t \in [t_0, t_{\max})$ :

$$
|\omega_1(x_1(t))| < M. \tag{D.93}
$$
Un résultat similaire peut être déduit pour $\omega_2^a(x_2)$. En effet, d'après (D.50), (D.51), (D.80), (D.82), et (D.93), pour tout $t \in [t_0, t_{\max})$:

$$
\begin{aligned}
|\omega_2^a(x_2(t))| & \le \max \left\{ \beta_2(\bar{\rho}_2(x_2(t_0)),0), \gamma_2^{\omega_1}(\beta_1(\rho_1(x_1(t_0)),0)), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\|\omega_2^a(x_2)\|_{[t_0,t)})), \right. \\
& \qquad \gamma_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t)})), \gamma_2^u(\|u_2\|_{[t_0,t)}), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\bar{\rho}_2(x_2(t_0))))) \\
& \qquad \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_2}(\|\omega_2^a(x_2)\|_{[t_0,t)}))), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,t)}))), \\
& \qquad \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(\|u_2\|_{[t_0,t)}))) \biggr\} \\
& \le \max \left\{ \beta_2(\bar{\rho}_2(x_2(t_0)),0), \gamma_2^{\omega_1}(\beta_1(\rho_1(x_1(t_0)),0)), \gamma_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t)})), \right. \\
& \qquad \gamma_2^u(\|u_2\|_{[t_0,t)}), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\bar{\rho}_2(x_2(t_0))))), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(M))), \\
& \qquad \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(\|u_2\|_{[t_0,t)}))), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\|\omega_2^a(x_2)\|_{[t_0,t)})), \\
& \qquad \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_2}(\|\omega_2^a(x_2)\|_{[t_0,t)}))) \biggr\}.
\end{aligned}
$$

Alors, en procédant par l'absurde et en utilisant (D.84) et (D.88), on déduit que

$$ |\omega_2^a(x_2(t))| \le M. \qquad (D.94) $$

Ainsi, en invoquant (D.51), (D.83), (D.81), (D.93) et (D.94), on peut prouver que $t_{\max} = \infty$ (par l'absurde), que le système (D.46)-(D.47) est bien uniformément positivement continu et que les états du système (D.46)-(D.47) sont bornés : (D.86) est satisfait.

### Etape 2. Propriété de stabilité (D.87).

Nous allons démontrer que les conditions de la Proposition D.1.3 sont satisfaites. Premièrement, on remarque, d'après (D.84), que :

$$
\left\{
\begin{array}{ll}
\max \left\{ \gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1^a}(s), \gamma_1^{\omega_2^b} \circ \eta_2^{\omega_2^a} \circ \gamma_2^{\omega_1^a}(s), \gamma_1^{\omega_2^b} \circ \eta_2^{\omega_1^a}(s) \right\} & \\
\le \max \left\{ \gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1^a}(m), \gamma_1^{\omega_2^b} \circ \eta_2^{\omega_2^a} \circ \gamma_2^{\omega_1^a}(m), \gamma_1^{\omega_2^b} \circ \eta_2^{\omega_1^a}(m) \right\} < m & \text{si } s \in [0,m) \\
\\
\max \left\{ \gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1^a}(s), \gamma_1^{\omega_2^b} \circ \eta_2^{\omega_2^a} \circ \gamma_2^{\omega_1^a}(s), \gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(s)) \right\} < s & \text{si } s \in [m,M].
\end{array}
\right.
\qquad (D.95)
$$
Ainsi, pour $t \in [t_0, \infty)$, on obtient, d'après (D.80), (D.82), (D.83), (D.93) et (D.95) :

$$
\begin{align*}
|\omega_1(x_1(t))| &\le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, t-t_0), \gamma_1^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0))|, 0)), \right. \\
&\qquad \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t]})), \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]})), \\
&\qquad \gamma_1^u(\|u_1\|_{[t_0,t]}), \gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|)), \\
&\qquad \gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\beta_2(|\omega_2(x_2(t_0))|, 0))), \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t]}))), \\
&\qquad \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]}))), \gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t]})), \\
&\qquad \left. \gamma_1^{\omega_2^b}(\eta_2^u(\|u_2\|_{[t_0,t]})) \right\} \\
\|\omega_1(x_1)\|_{[t_0,t]} &\le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, 0), \gamma_1^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0))|, 0)), \gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]})), \right. \\
&\qquad \gamma_1^u(\|u_1\|_{[t_0,t]}), \gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|)), \gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\beta_2(|\omega_2(x_2(t_0))|, 0))), \\
&\qquad \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t]}))), \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]}))), m \right\}, \tag{D.96}
\end{align*}
$$

par analogie, d'après (D.80), (D.82), (D.83), (D.84) et (D.96),

$$
\begin{align*}
\|\omega_2^a(x_2)\|_{[t_0,t)} &\le \max \left\{ \beta_2(|\omega_2(x_2(t_0))|, 0), \gamma_2^{\omega_1^1}(\beta_1(|\omega_1(x_1(t_0))|, 0)), \right. \\
&\qquad \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\|\omega_2^a(x_2)\|_{[t_0,t]})), \\
&\qquad \gamma_2^{\omega_1^1}(\gamma_1^u(\|u_1\|_{[t_0,t]})), \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|))), \\
&\qquad \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\|\omega_2^a(x_2)\|_{[t_0,t]}))), \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t]}))), \\
&\qquad \left. \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\|u_2\|_{[t_0,t]}))), \gamma_2^u(\|u_2\|_{[t_0,t]}) \right\} \\
&\le \max \left\{ \beta_2(\omega_2(|x_2(t_0))|, 0), \gamma_2^{\omega_1^1}(\beta_1(|\omega_1(x_1(t_0))|, 0)), \gamma_2^{\omega_1^1}(\gamma_1^u(\|u_1\|_{[t_0,t]})), \right. \\
&\qquad \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|))), \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\beta_1(|\omega_1(x_1(t_0))|, 0))), \\
&\qquad \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\|\omega_2^a(x_2(t_0))|, 0))), \\
&\qquad \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\|u_2\|_{[t_0,t]}))), \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\|u_2\|_{[t_0,t]}))), \\
&\qquad \left. \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0))|, 0))))), \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\|u_2\|_{[t_0,t]}))), \right. \\
&\qquad \left. \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\|u_2\|_{[t_0,t]}))))), \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(m))), \right. \\
&\qquad \left. \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^u}(\|u_2\|_{[t_0,t]}))), \gamma_2^u(\|u_2\|_{[t_0,t]}), m \right\}. \tag{D.97}
\end{align*}
$$
En incorporant les bornes de (D.96) et (D.97) dans (D.83), on déduit :

$$
\begin{equation}
\begin{aligned}
|\omega_2^b(x_2(t))| \le \max & \left\{ \alpha_2(|\omega_2^b(x_2(t_0))|), \eta_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0))|,0)), \right. \\
& \qquad \eta_2^{\omega_1}(\gamma_1^a(\beta_2(|\omega_2(x_2(t_0))|,0))), \eta_2^{\omega_1}(\gamma_1^a(\gamma_2^u(\|u_2\|_{[t_0,t]}))), \\
& \qquad \eta_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]})), \eta_2^{\omega_1}(\gamma_1^b(\alpha_2(|\omega_2^b(x_2(t_0))|))), \\
& \qquad \eta_2^{\omega_1}(\gamma_1^b(\eta_2^a(\beta_2(|\omega_2(x_2(t_0))|,0)))), \eta_2^{\omega_1}(\gamma_1^b(\eta_2^a(\gamma_2^u(\|u_2\|_{[t_0,t]})))), \\
& \qquad \eta_2^{\omega_1}(\gamma_1^b(\eta_2^u(\|u_2\|_{[t_0,t]}))), \eta_2^{\omega_1}(m), \eta_2^{\omega_2}(\beta_2(|\omega_2(x_2(t_0))|,0)), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0))|,0))), \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]}))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\alpha_2(|\omega_2^b(x_2(t_0))|))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\eta_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0))|,0))))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\eta_2^{\omega_1}(\gamma_1^a(\beta_2(|\omega_2(x_2(t_0))|,0)))))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\eta_2^{\omega_1}(\gamma_1^a(\gamma_2^u(\|u_2\|_{[t_0,t]})))))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\eta_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]}))))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\eta_2^{\omega_1}(\gamma_1^b(\alpha_2(|\omega_2^b(x_2(t_0))|))))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\eta_2^{\omega_1}(\gamma_1^b(\eta_2^a(\beta_2(|\omega_2(x_2(t_0))|,0)))))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\eta_2^{\omega_1}(\gamma_1^b(\eta_2^a(\gamma_2^u(\|u_2\|_{[t_0,t]})))))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\eta_2^{\omega_1}(\gamma_1^b(\eta_2^u(\|u_2\|_{[t_0,t]}))))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\eta_2^{\omega_1}(m))), \eta_2^{\omega_2}(\gamma_2^{\omega_1}(\gamma_1^b(\eta_2^u(\|u_2\|_{[t_0,t]})))), \\
& \qquad \eta_2^{\omega_2}(\gamma_2^u(\|u_2\|_{[t_0,t]})), \eta_2^a(m), \eta_2^u(\|u_2\|_{[t_0,t]}).
\end{aligned}
\tag{D.98}
\end{equation}
$$

Dans la suite, en remarquant que pour tout $x_2 \in \mathbb{R}^{n_{x_2}}$, d'après la Proposition A.1.2,

$$
|\omega_2(x_2)| \le \max \{2\omega_2^a(x_2), 2\omega_2^b(x_2)\}, \quad (D.99)
$$

on prouvera que $\omega_2(x_2)$ est borné en utilisant (D.97) et (D.98).

D'autre part, d'après (D.82) et (D.83), pour tout $t \in [t_0, t_{\max})$ :

$$
|\omega_2^a(x_2(t))| \le \max \left\{ \beta_2\left(|\omega_2\left(x_2\left(\frac{t}{2}\right)\right)|, \frac{t}{2}\right), \gamma_2^{\omega_1}\left(\|\omega_1(x_1)\|_{[\frac{t}{2},t)}\right), \gamma_2^u\left(\|u_2\|_{[\frac{t}{2},t)}\right) \right\}, \quad (\text{D.100})
$$

$$
\begin{equation}
\begin{aligned}
|\omega_2^b(x_2(t))| \le & \max \left\{ \alpha_2(|\omega_2^b(x_2(\frac{t}{2})))|, \eta_2^{\omega_1}(\|\omega_1(x_1)\|_{[\frac{t}{2},t)}), \eta_2^{\omega_2}(\|\omega_2^a(x_2)\|_{[\frac{t}{2},t)}) \right. \\
& \qquad \left. \eta_2^u(\|u_2\|_{[\frac{t}{2},t)}) \right\}.
\end{aligned}
\tag{D.101}
\end{equation}
$$
Par conséquent, d'après (D.80), (D.97), (D.98), (D.99), (D.100) et (D.101) :

$$
\begin{align*}
|\omega_1(x_1(t))| &\le \max \left\{ \beta_1(|\omega_1(x_1(\frac{t}{2}))|, \frac{t}{2}), \gamma_1^{\omega_2^a} (\|\omega_2^a(x_2)\|_{[\frac{t}{2},t)}), \gamma_1^{\omega_2^b} (\|\omega_2^b(x_2)\|_{[\frac{t}{2},t)}), \right. \\
&\qquad \left. \gamma_1^u (\|u_1\|_{[\frac{t}{2},t)}) \right\} \tag{D.102} \\
&\le \max \left\{ \beta_1(\beta_1(|\omega_1(x_1(t_0))|,0), \frac{t}{2}), \beta_1(\gamma_1^{\omega_2^a} (\beta_2(|\omega_2(x_2(t_0))|,0)), \frac{t}{2}), \right. \\
&\qquad \left. \beta_1(\gamma_1^{\omega_2^a} (\gamma_2^u (\|u_2\|_{[t_0,t)})), \frac{t}{2}), \beta_1(\gamma_1^u (\|u_1\|_{[t_0,t)}), \frac{t}{2}), \right. \\
&\qquad \left. \beta_1(\gamma_1^{\omega_2^b} (\alpha_2(|\omega_2^b(x_2(t_0))|)), \frac{t}{2}), \right. \\
&\qquad \left. \beta_1(\gamma_1^{\omega_2^b} (\eta_2^{\omega_2^a} (\beta_2(|\omega_2(x_2(t_0))|,0))), \frac{t}{2}), \right. \\
&\qquad \left. \beta_1(\gamma_1^{\omega_2^b} (\eta_2^{\omega_2^a} (\gamma_2^u (\|u_2\|_{[t_0,t)}))), \frac{t}{2}), \right. \\
&\qquad \left. \beta_1(\gamma_1^{\omega_2^b} (\eta_2^u (\|u_2\|_{[t_0,t)})), \frac{t}{2}), \beta_1(m, \frac{t}{2}), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\beta_2(|\omega_2(|x_2(t_0))|,0), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\beta_1(|\omega_1(x_1(t_0))|,0)), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^u (\|u_1\|_{[t_0,t)})), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^{\omega_2^b} (\alpha_2(|\omega_2^b(x_2(t_0))|))), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^{\omega_2^b} (\eta_2^{\omega_2^u} (\beta_1(|\omega_1(x_1(t_0))|,0)))), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^{\omega_2^b} (\eta_2^{\omega_2^u} (\gamma_1^{\omega_2^a} (\beta_2(|\omega_2(x_2(t_0))|,0))))), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^{\omega_2^b} (\eta_2^{\omega_2^u} (\gamma_1^{\omega_2^a} (\gamma_2^u (\|u_2\|_{[t_0,t)}))))), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^{\omega_2^b} (\eta_2^{\omega_2^u} (\gamma_1^{\omega_2^a} (\eta_2^u (\|u_2\|_{[t_0,t)}))))), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^{\omega_2^b} (\eta_2^{\omega_2^u} (\gamma_1^{\omega_2^a} (\eta_2^u (\|u_2\|_{[t_0,t)}))))), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^{\omega_2^b} (\eta_2^{\omega_2^u} (m))), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^{\omega_2^b} (\eta_2^u (\|u_2\|_{[t_0,t)}))), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^{\omega_2^b} (\eta_2^u (\|u_2\|_{[t_0,t)}))), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\gamma_2^u (\gamma_1^{\omega_2^b} (\eta_2^u (\|u_2\|_{[t_0,t)}))), \frac{t}{2})), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a} (\beta_2(2\eta_2^u (\beta_1(|\omega_1(x_1(t_0))|,0)), \frac{t}{2})), \right\}
\end{align*}
$$
$$
\begin{align*}
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^a}(\gamma_2^u(||u_2||_{[t_0,t]}))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^u(||u_1||_{[t_0,t]}))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(||u_2||_{[t_0,t]}))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^u(||u_2||_{[t_0,t]}))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(m), \frac{t}{2})), \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\beta_2(|\omega_2(x_2(t_0)|,0)), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_2^1(|\omega_1(x_1(t_0)|,0))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_2^u(||u_1||_{[t_0,t]}))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\beta_1(|\omega_1(x_1(t_0)|,0)))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\eta_1^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0)))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^a}(\gamma_2^u(||u_2||_{[t_0,t]}))))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\gamma_1^u(||u_1||_{[t_0,t]}))))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|))))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0))))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(||u_2||_{[t_0,t]}))))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(||u_2||_{[t_0,t]}))))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^a}(m))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^{\omega_2^b}(\eta_2^u(||u_2||_{[t_0,t]}))), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_1^u(||u_2||_{[t_0,t]}))), \frac{t}{2})), \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(m), \frac{t}{2})), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(||u_2||_{[t_0,t]})), \frac{t}{2})), \gamma_1^{\omega_2^a}(\gamma_2^{\omega_1^a}(||\omega_1(x_1)||_{[t/2,t]})), \\
& \gamma_1^{\omega_2^a}(\gamma_2^u(||u_2||_{[t/2,t]})),
\end{align*}
$$
$\gamma_1^{\omega_2^b}(\alpha_2(\alpha_2(|\omega_2^b(x_2(t_0))|))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\beta_1(|\omega_1(x_1(t_0)|,0))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]}))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\gamma_1^u(\|u_1\|_{[t_0,t]}))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]}))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(\|u_2\|_{[t_0,t]}))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(m))),\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0))),$
$\gamma_1^{\omega_2^b}(\alpha_2((\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\beta_1(|\omega_1(x_1(t_0)|,0))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^u(\|u_1\|_{[t_0,t]}))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_1(|\omega_1(x_1(t_0)|,0))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_1^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0)))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]}))))) ),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_1^u(\|u_1\|_{[t_0,t]}))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_1^{\omega_2^a}(\alpha_2(|\omega_2^b(x_2(t_0))|))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_1^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0)))))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]}))))) ),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]}))))) ),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(m)))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(\|u_2\|_{[t_0,t]}))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(\|u_2\|_{[t_0,t]}))),$
$\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(m))),\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\|u_2\|_{[t_0,t]}))),$
$\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\|\omega_1(x_1(t_0))\|_{[t_0,t]})),\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\beta_2(\omega_2(|x_2(t_0)|,0),\frac{t}{2}))),$
$\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\beta_1(|\omega_1(x_1(t_0)|,0)),\frac{t}{2}))),$
$\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^u(\|u_1\|_{[t_0,t]})),\frac{t}{2}))),$
$\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|)),\frac{t}{2}))),$
$$
\begin{align*}
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0)|,0)),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0)),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]})),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0)),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]})),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^u}(\|u_2\|_{[t_0,t]})),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(m)),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2m,\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\alpha_2(|\omega_2^b(x_2(t_0))|),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0)|,0)),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0)),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]})),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]})),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\alpha_2(|\omega_2^b(x_2(t_0))|),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(|\omega_2(x_2(t_0)|,0)),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]})),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^u}(\|u_2\|_{[t_0,t]})),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(m),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\beta_2(|\omega_2(x_2(t_0)|,0)),\frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\beta_1(|\omega_1(x_1(t_0)|,0)),\frac{t}{2}))),
\end{align*}
$$
où, pour $(s,t) \in \mathbb{R}_{\ge 0}^2$,

$$
\begin{aligned}
\tilde{\beta}_1(s,t) = \max & \left\{ \beta_1(\beta_1(s,0), \frac{t}{2}), \beta_1(\gamma_1^{\omega_1^a}(\beta_2(s,0)), \frac{t}{2}), \beta_1(\gamma_1^{\omega_2^b}(\alpha_2(s)), \frac{t}{2}) \right. \\
& \left. \beta_1(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(s,0))), \frac{t}{2}), \gamma_1^{\omega_2^a}(\beta_2(2\beta_2(s,0)), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\beta_1(s,0)), \frac{t}{2})), \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\alpha_2(s))), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\beta_1(s,0))), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\beta_2(s,0)))), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(s,0))))), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\alpha_2(s)), \frac{t}{2}), \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^1}(\beta_1(s,0)), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\beta_2(s,0))), \frac{t}{2})), \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\alpha_2(s))), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(s,0))))), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\beta_2(s,0)), \frac{t}{2})), \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_2^{\omega_1^1}(\beta_1(s,0))), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_2^{\omega_1^1}(\eta_2^{\omega_1^1}(\beta_1(s,0)))), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_2^{\omega_1^1}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\alpha_2(s))))), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^a}(\gamma_2^{\omega_1^1}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(s,0))))), \frac{t}{2}) \right. \\
& \left. \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\beta_2(\bar{\rho}(s),0)), \frac{t}{2}))), \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\beta_1(s,0)), \frac{t}{2}))), \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\alpha_2(s))), \frac{t}{2}))) \right. \\
& \left. \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\alpha_2(s))), \frac{t}{2}))), \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\beta_1(s,0))))), \frac{t}{2}))) \right\}
\end{aligned}
$$
$$
\begin{equation}
\left.
\begin{aligned}
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\alpha_2(s), \frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\beta_1(s,0)), \frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\beta_2(s,0)), \frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\alpha_2(s)), \frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(s,0))), \frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\beta_1(s,0)), \frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_1^{\omega_2^b}(\alpha_2(s))), \frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_1^{\omega_2^a}(\beta_2(s,0))), \frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(s,0))), \frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_1^{\omega_2^b}(\alpha_2(s))), \frac{t}{2}))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_1^{\omega_2^a}(\eta_2^{\omega_2^a}(\beta_2(s,0))), \frac{t}{2}))),
\end{aligned}
\right\} \tag{D.105}
\end{equation}
$$

$$
\begin{align*}
\tilde{\sigma}_1^u(s) = \max & \left\{ \beta_1(\gamma_1^{\omega_2^a}(\gamma_2^u(s)), 0), \beta_1(\gamma_1^u(s), 0), \beta_1(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(s))), 0), \right. \\
& \beta_1(\gamma_1^{\omega_2^b}(\eta_2^u(s)), 0), \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^u(s)), 0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\gamma_2^u(s))))), 0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(s))))), 0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))), 0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))), 0)), \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^u(s), 0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\gamma_2^u(s))), 0)), \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^u(s)), 0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))), 0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))), 0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\gamma_2^u(s))), 0))
\end{align*}
$$
$$
\begin{align*}
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\gamma_2^u(s))))),0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^u(s))))),0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(s))))),0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))),0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))),0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^u}(s)),0)),\gamma_1^{\omega_2^a}(\beta_2(2\eta_2^u(s),0)), \\
& \gamma_1^{\omega_2^a}(\gamma_2^u(s)),\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^u(s)),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\gamma_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1^u}(s),0))),\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\gamma_2^u(s))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^1}(\gamma_1^u(s)),0))),\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^u}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^u}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^u}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^u}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^u}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1^u}(s),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(s))),\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\gamma_2^u(s)))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^u(s))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s)))),\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^u(s))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1^1}(\gamma_1^{\omega_2^b}(\eta_2^u(s))))),
\end{align*}
$$
$$
\left.
\begin{array}{l}
\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^u(s)))), \\
\gamma_1^{\omega_2^b}(\alpha_2(\eta_2^u(s))), \gamma_1^{\omega_2^b}(\eta_2^u(s)), \gamma_1^u(s)
\end{array}
\right\}, \qquad (D.106)
$$

$$
\tilde{\sigma}_1^m(s) = \max \left\{
\begin{aligned}
& \beta_1(s,0), \gamma_1^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_2^{\omega_2^b}(\eta_2^{\omega_1}(s))), 0)), \gamma_1^{\omega_2^a}(\beta_2(2s,0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(s),0)), \\
& \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(s))), 0)), \gamma_1^{\omega_2^a}(\beta_2(2\eta_2^{\omega_2^a}(s),0)), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(s))), 0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2s,0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_1}(s),0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(s))), 0))), \\
& \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(2\eta_2^{\omega_2^a}(s),0))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1}(s))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(s))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(s)))
\end{aligned}
\right\}, \qquad (D.107)
$$

$$
\tilde{\sigma}_1^\omega(s) = \max \left\{
\begin{aligned}
& \gamma_1^{\omega_2^b}(\alpha_2(\alpha_2(s))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1}(\beta_1(s,0)))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\beta_2(s,0))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\alpha_2(s))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(s,0))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\beta_2(s,0)))), \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\beta_1(s,0))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\alpha_2(s))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\beta_1(s,0)))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^a}(\beta_2(s,0)))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\alpha_2(s)))))), \\
& \gamma_1^{\omega_2^b}(\alpha_2(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\beta_2(s,0)))))))).
\end{aligned}
\right\} \quad (D.108)
$$

De la même manière, d'après (D.96), (D.98), (D.99), (D.100), (D.101) et (D.102),
$$
\begin{aligned}
|\omega_2^a(x_2(t))| & \le \max \left\{ \beta_2(|\omega_2(x_2(\frac{t}{2}))|, \frac{t}{2}), \gamma_2^{\omega_1}(\|\omega_1(x_1)\|_{[\frac{t}{2},t)}), \gamma_2^u(\|u_2\|_{[\frac{t}{2},t)}) \right\}, \\
& \le \max \left\{ \beta_2(2\beta_2(|\omega_2(x_2(t_0))|,0), \frac{t}{2}), \beta_2(2\gamma_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0))|,0)), \frac{t}{2}), \right. \\
& \qquad \beta_2(2\gamma_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]})), \frac{t}{2}), \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(|\omega_2^b(x_2(t_0))|))), \frac{t}{2}), \\
& \qquad \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0))|,0)))), \frac{t}{2}), \\
& \qquad \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\beta_2(|\omega_2(x_2(t_0))|,0))))), \frac{t}{2}), \\
& \qquad \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\|u_2\|_{[t_0,t]}))))), \frac{t}{2}), \\
& \qquad \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]}))))), \frac{t}{2}), \\
& \qquad \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(|\omega_2^b(x_2(t_0))|))))), \frac{t}{2}), \\
& \qquad \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_2}(\beta_2(|\omega_2(x_2(t_0))|,0))))), \frac{t}{2}), \\
& \qquad \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_2}(\|u_2\|_{[t_0,t]}))))), \frac{t}{2}), \\
& \qquad \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(\|u_2\|_{[t_0,t]}))))), \frac{t}{2}), \\
& \qquad \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(m))), \frac{t}{2}), \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(\|u_2\|_{[t_0,t]}))), \frac{t}{2}), \\
& \qquad \beta_2(2\gamma_2^u(\|u_2\|_{[t_0,t]})), \frac{t}{2}), \beta_2(2m, \frac{t}{2}), \beta_2(2\alpha_2(|\omega_2^b(x_2(t_0))|), \frac{t}{2}), \\
& \qquad \beta_2(2\eta_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0))|,0)), \frac{t}{2}), \\
& \qquad \beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\beta_2(|\omega_2(x_2(t_0))|,0))), \frac{t}{2}), \\
& \qquad \beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\gamma_2^u(\|u_2\|_{[t_0,t]}))), \frac{t}{2}), \\
& \qquad \beta_2(2\eta_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]})), \frac{t}{2}), \\
& \qquad \beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(|\omega_2^b(x_2(t_0))|))), \frac{t}{2}), \\
& \qquad \beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_2}(\beta_2(|\omega_2(x_2(t_0))|,0))))), \frac{t}{2}), \\
& \qquad \beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_2}(\gamma_2^u(\|u_2\|_{[t_0,t]}))))), \frac{t}{2}), \\
& \qquad \beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(\|u_2\|_{[t_0,t]}))), \frac{t}{2}), \\
& \qquad \beta_2(2\eta_2^{\omega_1}(m), \frac{t}{2}), \beta_2(2\eta_2^{\omega_1}(\beta_2(|\omega_2(x_2(t_0))|,0)), \frac{t}{2}), \\
& \qquad \beta_2(2\eta_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]})), \frac{t}{2}),
\end{aligned}
$$
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^u(\|u_1\|_{[t_0,t]}))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\alpha_2(|\omega_2^b(x_2(t_0))|))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0))|,0))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_a}(\beta_2(|\omega_2(x_2(t_0))|,0))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_a}(\gamma_2^u(\|u_2\|_{[t_0,t]}))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^u(\|u_1\|_{[t_0,t]}))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\alpha_2(|\omega_2^b(x_2(t_0))|))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_a}(\beta_2(|\omega_2(x_2(t_0))|,0))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_a}(\gamma_2^u(\|u_2\|_{[t_0,t]}))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_a}(\|u_2\|_{[t_0,t]}))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(m))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^u(\|u_2\|_{[t_0,t]}))))), \frac{t}{2})$,
$\beta_2(2\eta_2^{\omega_2^a}(\gamma_2^u(\|u_2\|_{[t_0,t]})), \frac{t}{2}), \beta_2(2\eta_2^{\omega_2^a}(m), \frac{t}{2})$,
$\beta_2(2\eta_2^u(\|u_2\|_{[t_0,t]})), \frac{t}{2}), \gamma_2^{\omega_1}(\beta_1(\beta_1(|\omega_1(x_1(t_0))|,0), \frac{t}{2}))$,
$\gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_a}(\beta_2(|\omega_2(x_2(t_0))|,0)), \frac{t}{2})), \gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_a}(\gamma_2^u(\|u_2\|_{[t_0,t]})), \frac{t}{2}))$,
$\gamma_2^{\omega_1}(\beta_1(\gamma_1^u(\|u_1\|_{[t_0,t]})), \frac{t}{2}))$,
$\gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_b}(\alpha_2(|\omega_2^b(x_2(t_0))|))), \frac{t}{2})),$
$\gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_b}(\eta_2^{\omega_a}(\beta_2(|\omega_2(x_2(t_0))|,0))), \frac{t}{2})),$
$\gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_b}(\eta_2^{\omega_a}(\gamma_2^u(\|u_2\|_{[t_0,t]}))), \frac{t}{2})),$
$\gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_b}(\eta_2^u(\|u_2\|_{[t_0,t]}))), \frac{t}{2})),$
$\gamma_2^{\omega_1}(\beta_1(m, \frac{t}{2})), \gamma_2^{\omega_1}(\gamma_1^{\omega_a}(\|\omega_2^a(x_2)\|_{[\frac{t}{2},t]})),$
$\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\alpha_2(\alpha_2(|\omega_2^b(x_2(t_0))|)))),$
$\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\alpha_2(\eta_2^{\omega_1}(\beta_1(|\omega_1(x_1(t_0))|,0)))),$
$\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_a}(\beta_2(|\omega_2(x_2(t_0))|,0)))))),$
$$
\begin{align*}
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_2^u(\|u_2\|_{[t_0,t]}))))), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(\|u_2\|_{[t_0,t]}))))) \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(m))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\|\omega_2^a(x_2)\|_{[t/2,t]}))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\|u_2\|_{[t/2,t]}))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\|u_1\|_{[t/2,t]}))), \\
& \gamma_2^{\omega_1}(\|u_2\|_{[t/2,t]})). \tag{D.109}
\end{align*}
$$

que l'on écrit sous la forme :

$$
\begin{equation}
\begin{aligned}
|\omega_2^a(x_2(t))| \le \max & \left\{ \tilde{\beta}_2(\max \{|\omega_1(x_1(t_0))|, |\omega_2(x_2(t_0))|\}, t) \right. \\
& \phantom{\left\{ \right.} \left. \tilde{\sigma}_2^u(\max \{\|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t]}\}), \right. \\
& \phantom{\left\{ \right.} \left. \tilde{\sigma}_2^m(m), \tilde{\sigma}_2^\omega(\max \{|\omega_1(x_1(t_0))|, |\omega_2(x_2(t_0))|\}), \right. \\
& \phantom{\left\{ \right.} \left. \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\|\omega_2^a(x_2)\|_{[t/2,t]})), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\|\omega_2^a(x_2)\|_{[t_0,t]}))) \right\},
\end{aligned}
\tag{D.110}
\end{equation}
$$

où, pour $(s,t) \in \mathbb{R}^2_{\ge 0}$,

$$
\begin{align*}
\tilde{\beta}_2(s,t) = \max & \left\{ \beta_2\left(2\beta_2(s,0), \frac{t}{2}\right), \beta_2\left(2\gamma_2^{\omega_1}(\beta_1(s,0)), \frac{t}{2}\right), \right. \\
& \beta_2\left(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(s))), \frac{t}{2}\right), \\
& \beta_2\left(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\beta_1(s,0)))), \frac{t}{2}\right), \\
& \beta_2\left(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\beta_2(s,0))))), \frac{t}{2}\right), \\
& \beta_2\left(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(s))))), \frac{t}{2}\right), \\
& \beta_2\left(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\beta_2(s,0)))))), \frac{t}{2}\right), \\
& \beta_2\left(2\alpha_2(s), \frac{t}{2}\right), \beta_2\left(2\eta_2^{\omega_1}(\beta_1(s,0)), \frac{t}{2}\right), \\
& \beta_2\left(2\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\beta_2(s,0))), \frac{t}{2}\right), \beta_2\left(2\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(s))), \frac{t}{2}\right),
\end{align*}
$$
$$
\begin{equation}
\left.
\begin{aligned}
& \beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_a}(\beta_2(s,0))), \frac{t}{2}), \\
& \beta_2(2\eta_2^{\omega_a}(\beta_2(s,0)), \frac{t}{2}), \beta_2(2\eta_2^{\omega_a}(\gamma_2^{\omega_1}(\beta_1(s,0))), \frac{t}{2}), \\
& \beta_2(2\eta_2^{\omega_a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\alpha_2(s))), \frac{t}{2}), \\
& \beta_2(2\eta_2^{\omega_a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\beta_1(s,0)))), \frac{t}{2}), \\
& \beta_2(2\eta_2^{\omega_a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_a}(\beta_2(s,0))))), \frac{t}{2}), \\
& \beta_2(2\eta_2^{\omega_a}(\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\alpha_2(s))))), \frac{t}{2}), \\
& \beta_2(2\eta_2^{\omega_a}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\eta_1^{\omega_b}(\eta_2^{\omega_a}(\beta_2(s,0))))), \frac{t}{2}), \\
& \gamma_2^{\omega_1}(\beta_1(\beta_1(s,0), \frac{t}{2})), \gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_a}(\beta_2(s,0)), \frac{t}{2})), \\
& \gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_b}(\alpha_2(s)), \frac{t}{2})), \\
& \gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_b}(\eta_2^{\omega_a}(\beta_2(s,0))), \frac{t}{2})),
\end{aligned}
\right\} \tag{D.111}
\end{equation}
$$

$$
\begin{align*}
\tilde{\sigma}_2^u(s) &= \max \left\{
\begin{aligned}
& \beta_2(2\gamma_2^{\omega_1}(\gamma_1^u(s)),0), \\
& \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_a}(\gamma_2^u(s))))),0), \\
& \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^u(s))))),0), \\
& \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_a}(\gamma_2^u(s)))))),0), \\
& \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^u(s))))),0), \beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\eta_2^u(s))))),0), \\
& \beta_2(2\gamma_2^{\omega_1}(\gamma_1^u(s)),0), \beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_a}(\gamma_2^u(s))),0), \\
& \beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^u(s))),0), \beta_2(2\eta_2^{\omega_a}(\gamma_1^u(s))),0), \\
& \beta_2(2\eta_2^{\omega_a}(\gamma_1^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_a}(\gamma_2^u(s)))))),0), \\
& \beta_2(2\eta_2^{\omega_a}(\gamma_1^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^u(s))))),0), \\
& \beta_2(2\eta_2^{\omega_a}(\gamma_1^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^u(s)))))),0), \\
& \beta_2(2\eta_2^{\omega_a}(\gamma_1^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^u(s)))))),0), \\
& \beta_2(2\eta_2^{\omega_a}(\gamma_1^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^{\omega_1}(\gamma_1^{\omega_b}(\eta_2^u(s)))))),0), \beta_2(2\eta_2^{\omega_a}(\gamma_1^{\omega_1}(\eta_2^u(s))),0), \beta_2(2\eta_2^u(s,0)), \\
& \gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_a}(\gamma_2^u(s)),0)), \gamma_2^{\omega_1}(\beta_1(\gamma_1^u(s,0))), \\
& \gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_b}(\eta_2^{\omega_a}(\gamma_2^u(s))),0)), \gamma_2^{\omega_1}(\beta_1(\gamma_1^{\omega_b}(\eta_2^u(s)),0)),
\end{aligned}
\right\}
\end{align*}
$$
$$
\begin{align*}
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\gamma_2^u(s))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^u(s))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s)))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s)))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^u(s))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\gamma_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^u(s)))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^u(s)))))), \tag{D.112}
\end{align*}
$$

$$
\tilde{\sigma}_2^m(s) = \max \left\{
\begin{array}{l}
\beta_2(2\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))), 0), \beta_2(2s, 0), \beta_2(2\eta_2^{\omega_1}(s), 0), \\
\qquad \beta_2(2\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))), 0), \beta_2(2\eta_2^{\omega_1}(s), 0), \\
\qquad \gamma_2^{\omega_1}(\beta_1(s, 0)), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(s))), \\
\qquad \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))), \\
\qquad \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s)))))), \\
\qquad \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s)))))), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))),
\end{array}
\right\}, \tag{D.113}
$$

$$
\tilde{\sigma}_2^\omega(s) = \max \left\{
\begin{aligned}
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\alpha_2(s)))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\beta_1(s,0))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\beta_2(s,0))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(s))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_1^{\omega_2}(\alpha_2(s)))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_1^{\omega_2}(\eta_2^{\omega_1}(\beta_2(s,0))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(s)))))),
\end{aligned}
\right.
$$
$$
\begin{align*}
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_2}(\gamma_1^{\omega_1}(\gamma_2^{\omega_2}(\alpha_2(s)))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_2}(\gamma_1^{\omega_1}(\gamma_2^{\omega_1}(\beta_1(s,0))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_2}(\gamma_1^{\omega_1}(\gamma_2^{\omega_1}(\eta_1^{\omega_2}(\beta_2(s,0))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_2}(\gamma_1^{\omega_1}(\gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(s))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(\eta_2^{\omega_2}(\gamma_1^{\omega_1}(\gamma_2^{\omega_1}(\eta_1^{\omega_2}(\beta_2(s,0))))))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\beta_1(s,0)))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\beta_2(s,0)))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\alpha_2(s)))), \\
& \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_1}(\gamma_1^{\omega_2}(\eta_2^{\omega_2}(\beta_2(s,0)))))). \tag{D.114}
\end{align*}
$$

Par conséquent, d'après (D.104) et (D.110),

$$
\begin{align*}
\max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max & \left\{ \tilde{\beta}(\max \{|\omega_1(x_1(t_0))|, |\omega_2(x_2(t_0))|\}, t-t_0), \right. \\
& \phantom{=} \left. \tilde{\sigma}^u(\max \{\|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)}\}), \tilde{\sigma}^m(m), \tilde{\sigma}^x(\Delta), \right. \\
& \phantom{=} \left. \gamma_1^{\omega_2}(\gamma_2^{\omega_1}(\max \{\|\omega_1(x_1)\|_{[t_0,t)}, \|\omega_2^a(x_2)\|_{[t_0,t)}\})), \right. \\
& \phantom{=} \left. \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\max \{\|\omega_1(x_1)\|_{[t_0,t)}, \|\omega_2^a(x_2)\|_{[t_0,t)}\})), \right. \\
& \phantom{=} \left. \gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\gamma_2^{\omega_1}(\max \{\|\omega_1(x_1)\|_{[t_0,t)}, \|\omega_2^a(x_2)\|_{[t_0,t)}\}))), \right. \\
& \phantom{=} \left. \gamma_1^{\omega_2^b}(\eta_2^{\omega_1}(\max \{\|\omega_1(x_1)\|_{[t_0,t)}, \|\omega_2^a(x_2)\|_{[t_0,t)}\})), \right. \\
& \phantom{=} \left. \gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\eta_2^{\omega_2^a}(\max \{\|\omega_1(x_1)\|_{[t_0,t)}, \|\omega_2^a(x_2)\|_{[t_0,t)}\}))) \right\} \tag{D.115}
\end{align*}
$$

où

$$
\tilde{\beta}(s,t) = \max \{\tilde{\beta}_1(s,t), \tilde{\beta}_2(s,t)\}, \tag{D.116}
$$

$$
\tilde{\sigma}^u(s) = \max \{\tilde{\sigma}_1^u(s), \tilde{\sigma}_2^u(s)\}, \tag{D.117}
$$

$$
\tilde{\sigma}^m(s) = \max \{\tilde{\sigma}_1^m(s), \tilde{\sigma}_2^m(s)\}, \tag{D.118}
$$

$$
\tilde{\sigma}^x(s) = \max \left\{ \tilde{\sigma}_1^\omega(\max \{\rho_1(s), \bar{\rho}_2(s)\}), \tilde{\sigma}_2^\omega(\max \{\rho_1(s), \bar{\rho}_2(s)\}) \right\}. \quad (\text{D.119})
$$

Soient, pour $s \in \mathbb{R}_{\ge 0}$,

$$
\sigma^y(s) = \max \{ \gamma_1^{\omega_2^a}(s), 2\gamma_2^{\omega_1^1}(s), 2\eta_2^{\omega_1^1}(s), 2\eta_2^{\omega_2^a}(s) \} \quad (\text{D.120})
$$

$$
\sigma^u(s) = \max \{\gamma_1^u(s), 2\gamma_2^u(s), 2\eta_2^u(s)\} \quad (\text{D.121})
$$

$$
\tilde{\gamma}^u(s) = \max \{\tilde{\beta}(\sigma^y(\tilde{\sigma}^u(s)), 0), \tilde{\beta}(\sigma^u(s), 0), \tilde{\sigma}^u(s)\}. \quad (\text{D.122})
$$

On peut dès lors voir que les hypothèses de la Proposition D.1.3 sont satisfaites. En effet,
en identifiant $x = \max \{|\omega_1(x_1)|, |\omega_2(x_2)|\}$, $y = \max \{|\omega_1(x_1)|, |\omega_2^a(x_2)|\}$, $\beta = \tilde{\beta}$, pour $s \in \mathbb{R}_{\ge 0}$,
$\gamma^y(s) = \max \{\gamma_1^{\omega_2^a} \circ \gamma_2^{\omega_1^1}(s), \gamma_2^{\omega_1^1} \circ \gamma_1^{\omega_2^a}(s), \gamma_1^{\omega_2^b} \circ \eta_2^{\omega_2^a} \circ \gamma_2^{\omega_1^1}(s), \gamma_1^{\omega_2^b} \circ \eta_2^{\omega_1^1}(s), \gamma_2^{\omega_1^1} \circ \gamma_1^{\omega_2^b} \circ \eta_2^{\omega_2^a}(s)\}$,
$\gamma^u(s) = \tilde{\sigma}^u(s)$, $c_y = \max\{\tilde{\sigma}^m(m), \tilde{\sigma}^x(\Delta)\}$, $\sigma^t = 0$, $\sigma^x = \max\{\beta_1(s,0), 2\beta_2(s,0), 2\alpha_2(s)\}$, $\sigma^y$ est donnée dans (D.120), $\sigma^u$ dans (D.121), $c_x = 0$, $c = \max\{c_y, m\}$, $\tilde{\gamma}^u$ dans (D.122), (D.25) est vérifiée d'après (D.115), (D.26) d'après (D.93) et (D.94), (D.27) d'après (D.80), (D.82), (D.83) et (D.99), (D.28) d'après (D.84), (D.29) d'après (D.85) et (D.88), (D.31) avec $\bar{\Delta} = \Delta$ puisque $\max\{\nu^x(\Delta), \nu^u(\Delta)\} < M$. Ainsi, puisque (D.88) est garantie, d'après la Proposition D.1.3, pour $\eta > 1$, il existe $\beta \in \mathcal{KL}$ telle que, pour tout $t \in [t_0, \infty)$ :

$$ \max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max \left\{ \begin{array}{l} \beta(\max \{|\omega_1(x_1(t_0))|, |\omega_2(x_2(t_0))|\}, t-t_0), \\ \tilde{\gamma}^u(\max \{\|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)}\}), \eta \tilde{\sigma}^m(m), \\ \eta \tilde{\sigma}^x(\Delta), \eta m \end{array} \right\}. \quad (D.123) $$

Puisque $\delta^m(s) = \max\{\eta \tilde{\sigma}^m(s), \eta s\}$ et $\delta^x(s) = \eta \tilde{\sigma}^x(s)$ pour $s \in \mathbb{R}_{\ge 0}$, le résultat désiré est obtenu. $\square$

## D.3 Théorèmes du petit gain pour des systèmes paramétrés

Les théorèmes de la section précédente sont étendus à la famille paramétrée de systèmes suivante :

$$ \left\{ \begin{array}{ll} \dot{x}_1 = f_{\tau,1}(t,x_1,x_2,u_1) & \forall t \in [t_{i-1},t_i] \\ x_1(t_i^+) = g_{\tau,1}(t_i,x_1(t_i),x_2(t_i),u_1(t_i)) & \end{array} \right. \qquad (D.124) $$

$$ \left\{ \begin{array}{ll} \dot{x}_2 = f_{\tau,2}(t,x_2,x_1,u_2) & \forall t \in [t_{i-1},t_i] \\ x_2(t_i^+) = g_{\tau,2}(t_i,x_2(t_i),x_1(t_i),u_2(t_i)), & \end{array} \right. \qquad (D.125) $$

qui correspondent aux systèmes (D.46)-(D.47) paramétrés en $\tau \in \mathbb{R}_{>0}$, aussi bien au niveau des dynamiques que des instants de sauts, puisque pour tout $i \in \mathbb{Z}_{>0}$, on rappelle que :

$$ v \le t_i - t_{i-1} \le \tau. \qquad (D.126) $$

Lorsque la classe de systèmes n'est pas affectée par des sauts, on peut remplacer $v$ par 0.

**Théorème D.3.1** On suppose qu'il existe $\beta_1, \beta_2 \in \mathcal{KL}$, $\alpha_1, \alpha_2, \gamma_2^{\omega_1}, \gamma_2^u, \eta_1^t, \eta_2^t, \eta_1^{\omega_1}, \eta_2^{\omega_2}, \eta_1^u \in \mathcal{K}$, $\gamma_1^{\omega_2}, \gamma_1^u, \eta_1^{\omega_1}, \eta_1^{\omega_2}, \eta_1^u \in \mathcal{KK}$, $\bar{\tau} \in [v, \infty]$, tels que pour tout $x_1(t_0) \in \mathbb{R}^{n_{x_1}}$, $(u_1, x_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{x_2}}$, $\tau \in [v, \bar{\tau})$, $t \ge t_0 \ge 0$, les solutions de (D.124) vérifient :

$$ |\omega_1(x_1(t))| \le \max \left\{ \begin{array}{l} \beta_1(|\omega_1(x_1(t_0))|, t - t_0), \gamma_1^{\omega_2}(\tau, \|\omega_2^a(x_2)\|_{[t_0,t)}), \\ \gamma_1^u(\tau, \|u_1\|_{[t_0,t)}) \end{array} \right\} \qquad (D.127) $$

$$ |x_1(t)| \le \max \left\{ \begin{array}{l} \alpha_1(|x_1(t_0)|), \eta_1^t(t - t_0), \eta_1^{\omega_1}(\tau, \|\omega_1(x_1)\|_{[t_0,t)}), \eta_1^{\omega_2}(\tau, \|\omega_2^a(x_2)\|_{[t_0,t)}), \\ \eta_1^{\omega_2}(\tau, \|\omega_2^b(x_2)\|_{[t_0,t)}), \eta_1^u(\tau, \|u_1\|_{[t_0,t)}) \end{array} \right\}, \qquad (D.128) $$
pour tout $x_2(t_0) \in \mathbb{R}^{n_{x_2}}$, $(u_2,x_1) \in \mathcal{L}_{\infty}^{n_{u_2}+n_{x_1}}$, $\tau \in [v,\bar{\tau})$, $t \ge t_0 \ge 0$, les solutions de (D.125) vérifient :

$$|\omega_2^a(x_2(t))| \le \max \{\beta_2(|\omega_2^a(x_2(t_0))|,t-t_0), \gamma_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,t)}), \gamma_2^u(\|u_2\|_{[t_0,t)})\} \quad (\text{D.129})$$

$$\begin{aligned} |\omega_2^b(x_2(t))| \le \max & \{\alpha_2(|\omega_2^b(x_2(t_0))|), \eta_2^t(t-t_0), \eta_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,t)}), \\ & \eta_2^{\omega_2}(\|\omega_2^a(x_2)\|_{[t_0,t)}), \eta_2^u(\|u_2\|_{[t_0,t)})\}, \end{aligned} \quad (\text{D.130})$$

et, pour tout $\Delta, \varepsilon \in \mathbb{R}_{>0}$, soit $M \in \mathbb{R}_{>0}$ (suffisamment grand) et $m \in \mathbb{R}_{>0}$ (suffisamment petit) tels que

$$\left\{ \begin{array}{l} \delta^m(\bar{\tau},m) < \varepsilon \\ \max \{m, \tilde{\sigma}^m(\bar{\tau},m), \nu^x(\bar{\tau},\Delta), \nu^u(\bar{\tau},\Delta)\} < M, \end{array} \right. \quad (\text{D.131})$$

où pour $(\tau,s) \in [v,\bar{\tau}) \times \mathbb{R}_{\ge 0}$,

$$\delta^m(\tau,s) = \max \{\eta \tilde{\sigma}^m(\tau,s), \eta s\}$$

$$\tilde{\sigma}^m(\tau,s) = \max \{\beta_1(s,0), \gamma_1^{\omega_2^a}(\tau, \beta_2(s,0)), \beta_2(s,0), \gamma_2^{\omega_1^1}(\beta_1(s,0))\}$$

$$\nu^x(\tau,s) = \max \{\beta_1(\rho_1(s),0), \gamma_1^{\omega_2^a}(\tau, \beta_2(\bar{\rho}_2(s),0)), \rho_1(s), \beta_2(\bar{\rho}_2(s),0), \gamma_2^{\omega_1^1}(\beta_1(\rho_1(s),0)) \\ \tilde{\beta}(s,0),s\}$$

$$\nu^u(\tau,s) = \max \{\gamma_1^u(\tau,s), \gamma_1^{\omega_2^a}(\tau, \gamma_2^u(s)), \gamma_2^u(s), \gamma_2^{\omega_1^1}(\gamma_1^u(\tau,s)), \tilde{\gamma}^u(\tau,s)\},$$

avec $\eta > 1$, $\tilde{\beta} \in \mathcal{KL}$, $\tilde{\gamma}^u \in \mathcal{KK}$ sont respectivement définis par (D.151) et (D.156). Il existe $\tau^*(\varepsilon,\Delta) \in [v,\bar{\tau})$ (défini dans (D.135)), que l'on note $\tau^*$, tel que

$$\max \{\gamma_1^{\omega_2^a}(\tau^*, \gamma_2^{\omega_1^1}(s)), \gamma_2^{\omega_1^1}(\gamma_1^{\omega_2^a}(\tau^*, s))\} < s \quad \text{pour } s \in [m,M], \quad (\text{D.132})$$

alors le système (D.124)-(D.125) est uniformément positivement complet pour tout $\tau \in [v,\tau^*)$, $(x_1(t_0),x_2(t_0)) \in \mathbb{R}^{n_{x_1}+n_{x_2}}$ et $(u_1,u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ avec $\max \{|x_1(t_0)|,|\omega_2^a(x_2(t_0))|, \|u_1\|_{\infty}, \|u_2\|_{\infty}\} < \Delta$. De plus, il existe $\beta \in \mathcal{KL}$, $\sigma \in \mathcal{K}$, $\bar{\sigma} \in \mathcal{KK}$, tels que pour tout $\tau \in [v,\tau^*)$, $(x_1(t_0),x_2(t_0)) \in \mathbb{R}^{n_{x_1}+n_{x_2}}$ et $(u_1,u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ où $\max \{|x_1(t_0)|,|\omega_2^a(x_2(t_0))|, \|u_1\|_{\infty}, \|u_2\|_{\infty}\} < \Delta$, les solutions de (D.124)-(D.125) satisfont, pour tout $t \ge t_0 \ge 0$,

$$\max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max \left\{ \begin{array}{l} \beta(\max \{|\omega_1(x_1(t_0))|, |\omega_2^a(x_2(t_0))|\}, t-t_0), \\ \sigma(\max \{\|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)}\}), \\ \bar{\sigma}(\tau^*, \max \{\|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)}\}), \varepsilon \end{array} \right\}. \quad (\text{D.133})$$

**Preuve.** Après avoir prouvé qu'il existe bien $\tau^*$ tel que (D.132) est satisfait, il est prouvé que le système est uniformément positivement complet lorsque les conditions initiales et les entrées $(u_1,u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ sont convenablement majorées. La propriété de stabilité désirée est ensuite obtenue en invoquant la Proposition D.1.3.

Soient $\Delta, \varepsilon, \bar{\theta} \in \mathbb{R}_{>0}$ et considérons $m, M \in \mathbb{R}_{>0}$ tels que (D.131) soit vérifié. On définit l'application bijective suivante

$$\begin{array}{rcl} \gamma: & [0,\tau_1) \times [0,s_1) & \longrightarrow [0,\tau_2) \times [0,s_2) \\ & (\tau,s) & \longmapsto \max \{\gamma_1^2(\tau,\gamma_2^1(s)), \gamma_2^1(\gamma_1^2(\tau,s))\} \end{array} \quad (\text{D.134})$$
où $\tau_1, \tau_2, s_1, s_2 \in \mathbb{R}_{>0} \cup \{\infty\}$. Soit $\gamma^{-1}$ l'inverse de $\gamma$ par rapport à $\tau$. On prend la convention que $\gamma^{-1}(s) = \infty$ si $s \ge \tau_2$. Ainsi, il suffit pour garantir (D.132) de prendre :

$$ \tau^* = \min \left\{ \bar{\tau}, \min_{s \in [m, M]} \gamma^{-1}(s) \right\}. \qquad (D.135) $$

*Etaple 1. Complétude positive et majoration de $\omega_1(x_1)$ et $\omega_2^a(x_2)$.*

Soient $t_0 \in \mathbb{R}_{\ge 0}$, $\tau \in [\upsilon, \tau^*)$, $(x_1(t_0), x_2(t_0)) \in \mathbb{R}^{n_{x_1}+n_{x_2}}$ et $(u_1, u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ tels que

$$ \max \{|x_1(t_0)|, |\omega_2^a(x_2(t_0))|, \|u_1\|_{\infty}, \|u_2\|_{\infty}\} < \Delta, \qquad (D.136) $$

ce qui implique, d'après la définition de $M$,

$$ \nu^x(\tau, \max \{|x_1(t_0)|, |\omega_2^a(x_2(t_0))|\}) < \nu^x(\bar{\tau}, \max \{|x_1(t_0)|, |\omega_2^a(x_2(t_0))|\}) \\ < M \qquad (D.137) $$

$$ \nu^u(\tau, \max \{ \|u_1\|_{[t_0, \infty)}, \|u_2\|_{[t_0, \infty)} \}) < \nu^u(\bar{\tau}, \max \{ \|u_1\|_{[t_0, \infty)}, \|u_2\|_{[t_0, \infty)} \}) \\ < M. \qquad (D.138) $$

Soit $[t_0, t_{\max})$ l'intervalle maximal d'existence du système (D.124)-(D.125), où $t_{\max} \in (t_0, \infty]$. De par (D.127) et (D.129), on a, pour tout $t \in [t_0, t_{\max})$ :

$$ |\omega_1(x_1(t))| \le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, t-t_0), \gamma_1^{\omega_2^a}(\tau, \beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \\ \qquad \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t]})), \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_2^u}(\|u_2\|_{[t_0,t]})), \\ \qquad \left. \gamma_1^{\omega_1^u}(\tau, \|u_1\|_{[t_0,t]}) \right\} \qquad (D.139) $$

$$ \|\omega_1(x_1)\|_{[t_0, t)} \le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, 0), \gamma_1^{\omega_2^a}(\tau, \beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \\ \qquad \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t]})), \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_2^u}(\|u_2\|_{[t_0,t]})), \\ \qquad \left. \gamma_1^{\omega_1^u}(\tau, \|u_1\|_{[t_0,t]}) \right\}. \qquad (D.140) $$

Nous allons montrer par l'absurde que $|\omega_1(x_1(t))| < M$ pour tout $t \in [t_0, t_{\max})$. Dans un premier temps, on remarque que $|\omega_1(x_1(t_0))| < M$, d'après (D.50) et (D.137). On suppose qu'il existe $t_1 \in (t_0, t_{\max})$ tel que $|\omega_1(x_1(t_1))| = M$. Soit $\bar{t} = \inf \{t \in (t_0, t_{\max}) : |\omega_1(x_1(t))| = M\}$. De par (D.140), on a :

$$ \begin{aligned} \|\omega_1(x_1)\|_{[t_0, \bar{t})} &= |\omega_1(x_1(\bar{t}))| \le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, 0), \gamma_1^{\omega_2^a}(\tau, \beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \\ & \qquad \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0, \bar{t})})), \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_2^u}(\|u_2\|_{[t_0, \bar{t})})), \\ & \qquad \left. \gamma_1^{\omega_1^u}(\tau, \|u_1\|_{[t_0, \bar{t})}) \right\} \\ &= \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, 0), \gamma_1^{\omega_2^a}(\tau, \beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \\ & \qquad \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_1^1}(M)), \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_2^u}(\|u_2\|_{[t_0, \bar{t})})), \\ & \qquad \left. \gamma_1^{\omega_1^u}(\tau, \|u_1\|_{[t_0, \bar{t})}) \right\}. \end{aligned} \qquad (D.141) $$
Notons que, d'après (D.137) et (D.138) :

$$
\max \left\{ \beta_1(|\omega_1(x_1(t_0))|,0), \gamma_1^{\omega_2^a}(\bar{\tau}, \beta_2(|\omega_2^a(x_2(t_0))|,0)), \gamma_1^{\omega_2^a}(\bar{\tau}, \gamma_2^u(\|u_2\|_{[t_0,t]})), \gamma_1^u(\bar{\tau}, \|u_1\|_{[t_0,t]}) \right\} < M, \quad (D.142)
$$

de plus, en vu de (D.132) :

$$
\gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_1^1}(M)) < \gamma_1^{\omega_2^a}(\bar{\tau}, \gamma_2^{\omega_1^1}(M)) < M, \quad (D.143)
$$

par conséquent, d'après (D.141), (D.142), (D.143),

$$
|\omega_1(x_1(\bar{t}))| < M
$$

ce qui contredit le fait que $|\omega_1(x_1(t))| = M$, ainsi puisque $\omega_1(x_1)$ est continue entre deux instants de saut et d'après (D.48), pour tout $t \in [t_0, t_{\max})$ :

$$
|\omega_1(x_1(t))| < M. \tag{D.144}
$$

Par analogie, on peut montrer que pour tout $t \in [t_0, t_{\max})$ :

$$
|\omega_2^a(x_2(t))| < M. \tag{D.145}
$$

En conséquence, en invoquant (D.51), (D.128), (D.130), (D.144) et (D.145), on obtient, par l'absurde, que $t_{\max} = \infty$ et on peut montrer que le système (D.124)-(D.125) est uniformément positivement complet.

Etape 2. Propriété (D.133).

Nous allons démontrer que les conditions de la Proposition D.1.3 sont vérifiées. Premièrement,
notons que d'après (D.132) et puisque $\gamma_1^{\omega_2^a}(\cdot, \gamma_2^{\omega_1^1}(\cdot)) \in \mathcal{K}\mathcal{K}$ :

$$
\left\{
\begin{array}{ll}
\gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_1^1}(s)) \le \gamma_1^{\omega_2^a}(\tau^*, \gamma_2^{\omega_1^1}(m)) < m & \text{si } s \in [0, m) \\
\gamma_1^{\omega_2^a}(\tau^*, \gamma_2^{\omega_1^1}(s)) < s & \text{si } s \in [m, M].
\end{array}
\right.
\qquad (D.146)
$$

Ainsi, pour $t \in [t_0, \infty)$, on a, d'après (D.139), (D.144) et (D.146) :

$$
\begin{align*}
|\omega_1(x_1(t))| &\le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, t-t_0), \gamma_1^{\omega_2^a}(\tau, \beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_1^1}(\|\omega_1(x_1)\|_{[t_0,t]})), \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_1^u}(\|u_2\|_{[t_0,t]})), \gamma_1^{\omega_1^u}(\tau, \|u_1\|_{[t_0,t]}) \right\} \\
\|\omega_1(x_1)\|_{[t_0,t]} &\le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, 0), \gamma_1^{\omega_2^a}(\tau, \beta_2(|\omega_2^a(x_2(t_0))|, 0)), \right. \\
&\qquad \left. \gamma_1^{\omega_2^a}(\tau, \gamma_2^{\omega_1^u}(\|u_2\|_{[t_0,t]})), \gamma_1^{\omega_1^u}(\tau, \|u_1\|_{[t_0,t]}), m \right\}, \tag{D.147}
\end{align*}
$$

par symétrie,

$$
\|\omega_2^a(x_2)\|_{[t_0,t)} \le \max \left\{ \beta_2(|\omega_2^a(x_2(t_0))|,0), \gamma_2^{\omega_1^1}(\beta_1(|\omega_1(x_1(t_0))|,0)), \right. \\
\qquad \left. \gamma_2^{\omega_1^1}(\gamma_1^u(\tau, \|u_1\|_{[t_0,t]})), \gamma_2^u(\|u_2\|_{[t_0,t]}), m \right\}. \quad (D.148)
$$
D'autre part, d'après (D.129), pour tout $t \in [t_0, t_{\max})$ :

$$|\omega_2^a(x_2(t))| \le \max \left\{ \beta_2\left(|\omega_2^a(x_2(\frac{t}{2}))|, \frac{t}{2}\right), \gamma_2^u\left(\|\omega_1(x_1)\|_{[\frac{t}{2},t)}\right), \gamma_2^u\left(\|u_2\|_{[\frac{t}{2},t)}\right) \right\}, \quad (\text{D.149})$$

par conséquent, d'après (D.127), (D.147), (D.148) et (D.149) :

$$\begin{aligned} |\omega_1(x_1(t))| &\le \max \left\{ \beta_1\left(|\omega_1(x_1(\frac{t}{2}))|, \frac{t}{2}\right), \gamma_1^a\left(\tau, \|\omega_2^a(x_2)\|_{[\frac{t}{2},t)}\right), \gamma_1^u\left(\tau, \|u_1\|_{[\frac{t}{2},t)}\right) \right\} \\ &\le \max \left\{ \beta_1\left(\beta_1\left(|\omega_1(x_1(t_0))|, 0\right), \frac{t}{2}\right), \beta_1\left(\gamma_1^a\left(\tau, \beta_2\left(|\omega_2^a(x_2(t_0))|, 0\right)\right), \frac{t}{2}\right), \right. \\ &\qquad \beta_1\left(\gamma_1^a\left(\tau, \gamma_2^u\left(\|u_2\|_{[t_0,t)}\right)\right), \frac{t}{2}\right), \beta_1\left(\gamma_1^u\left(\tau, \|u_1\|_{[t_0,t)}\right), \frac{t}{2}\right), \\ &\qquad \beta_1\left(m, \frac{t}{2}\right), \gamma_1^a\left(\tau, \beta_2\left(\beta_2\left(|\omega_2^a(x_2(t_0))|, 0\right)\right), \frac{t}{2}\right), \\ &\qquad \gamma_1^a\left(\tau, \beta_2\left(\gamma_2^u\left(\beta_1\left(|\omega_1(x_1(t_0))|, 0\right)\right), \frac{t}{2}\right)\right), \\ &\qquad \gamma_1^a\left(\tau, \beta_2\left(\gamma_2^u\left(\gamma_1^u\left(\tau, \|u_1\|_{[t_0,t)}\right)\right), \frac{t}{2}\right)\right), \gamma_1^a\left(\tau, \beta_2\left(\gamma_2^u\left(\|u_2\|_{[t_0,t)}\right), \frac{t}{2}\right)\right), \\ &\qquad \gamma_1^a\left(\tau, \beta_2\left(m, \frac{t}{2}\right)\right), \gamma_1^a\left(\tau, \gamma_2^u\left(\|\omega_1(x_1)\|_{[\frac{t}{2},t)}\right)\right), \gamma_1^a\left(\tau, \gamma_2^u\left(\|u_2\|_{[\frac{t}{2},t)}\right)\right), \\ &\qquad \gamma_1^u\left(\tau, \|u_1\|_{[\frac{t}{2},t)}\right) \end{aligned}$$

un résultat analogue est vérifié par $\omega_2^a(x_2)$, ainsi,

$$\max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max \left\{ \tilde{\beta}\left(\max \{|\omega_1(x_1(t_0))|, |\omega_2^a(x_2(t_0))|\}, t - t_0\right), \right. \\ \sigma^u\left(\tau, \max \left\{ \|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)} \right\}\right), \tilde{\sigma}^m\left(\tau, m\right), \\ \gamma_1^a\left(\tau, \gamma_2^u\left(\|\omega_1(x_1)\|_{[0,t)}\right)\right), \\ \gamma_2^u\left(\gamma_1^a\left(\tau, \|\omega_1(x_1)\|_{[0,t)}\right)\right) \left\} \right\}, \quad (\text{D.150})$$

où

$$\tilde{\beta}(s,t) = \max \left\{ \beta_1\left(\beta_1(s,0), \frac{t}{2}\right), \beta_1\left(\gamma_1^a\left(\tau^*, \beta_2(s,0)\right), \frac{t}{2}\right), \gamma_1^a\left(\tau^*, \beta_2\left(\beta_2(s,0), \frac{t}{2}\right)\right), \right. \\ \gamma_1^a\left(\tau^*, \beta_2\left(\gamma_2^u\left(\beta_1(s,0), \frac{t}{2}\right)\right) \right. \\ \left. \beta_2\left(\beta_2(s,0), \frac{t}{2}\right), \beta_2\left(\gamma_2^u\left(\beta_1(s,0), \frac{t}{2}\right)\right), \gamma_2^u\left(\beta_1\left(\beta_1(s,0), \frac{t}{2}\right)\right), \right. \\ \left. \gamma_2^u\left(\beta_1\left(\gamma_1^a\left(\tau^*, \beta_2(s,0), \frac{t}{2}\right)\right)\right) \right\}, \quad (\text{D.151})$$

$$\sigma^u(\tau,s) = \max\{\bar{\sigma}^u(\tau,s), \tilde{\sigma}^u(s)\}, \quad (\text{D.152})$$

$$\begin{aligned} \bar{\sigma}^u(\tau,s) = \max & \left\{ \beta_1(\gamma_1^a(\tau, \gamma_2^u(s)), 0), \beta_1(\gamma_1^u(\tau, s), 0), \gamma_1^a(\tau, \beta_2(\gamma_2^u(\gamma_1^u(\tau, s)), 0)), \right. \\ & \gamma_1^a(\tau, \beta_2(\gamma_2^u(s), 0)), \gamma_1^a(\tau, \gamma_2^u(s)), \gamma_1^u(\tau, s), \\ & \beta_2(\gamma_2^u(\gamma_1^u(\tau, s)), 0), \gamma_2^u(\beta_1(\gamma_1^a(\tau, \gamma_2^u(s)), 0)), \\ & \gamma_2^u(\beta_1(\gamma_1^u(\tau, s), 0)), \gamma_2^u(\gamma_1^u(\tau, s)) \right\}, \\ \tilde{\sigma}^u(s) = \max & \left\{ \beta_2(\gamma_2^u(s), 0), \gamma_2^u(s) \right\}, \end{aligned} \quad (\text{D.153})$$
$$ \tilde{\sigma}^m(\tau, s) = \max \{ \beta_1(s,0), \gamma_1^{\omega_2}(\tau, \beta_2(s,0)), \beta_2(s,0), \gamma_2^{\omega_1}(\beta_1(s,0)) \}. \quad (D.154) $$

Soit, pour $s \in \mathbb{R}_{\ge 0}$,

$$ \sigma^y(\tau, s) = \max \{ \gamma_1^{\omega_2}(\tau, \gamma_2^{\omega_1}(s)), \gamma_2^{\omega_1}(\tau, s) \} \quad (D.155) $$

$$ \tilde{\gamma}^u(\tau, s) = \max \{ \tilde{\beta}(\sigma^y(\tau, \sigma^u(\tau, s), 0)), \tilde{\beta}(\sigma^u(\tau, s), 0), \sigma^u(\tau, s) \}. \quad (D.156) $$

On constate que les conditions de la Proposition D.1.3 sont satisfaites. En effet, on identifie $x = y = \max \{|\omega_1(x_1)|, |\omega_2^a(x_2)|\}$, $u = \max\{|u_1|, |u_2|\}$, pour $(s,t) \in \mathbb{R}_{\ge 0}^2$, $\beta(s,t) = \tilde{\beta}(s,t)$, $\gamma^y(s) = \max \{\gamma_1^{\omega_2}(\tau^*, \gamma_2^{\omega_1}(s)), \gamma_2^{\omega_1}(\gamma_1^{\omega_2}(\tau^*, s))\}$, $\gamma^u(s) = \sigma^u(\tau^*, s)$, $c_y = \tilde{\sigma}^m(\tau^*, m)$, $\sigma^t(s) = 0$, $\sigma^x(s) = \tilde{\beta}(s,0)$, $\sigma^y$ est défini dans (D.155) avec $\tau = \tau^*$, $\sigma^u(s) = \sigma^u(\tau^*, s)$, $c_x = \tilde{\sigma}^m(\tau^*, m)$, $c = \max \{\tilde{\sigma}^m(\tau^*, m), m\}$, $\tilde{\gamma}^u$ est donné dans (D.156) avec $\tau = \tau^*$. L'inégalité (D.25) est satisfaite d'après (D.150), (D.26) d'après (D.144) et (D.145), (D.27) d'après (D.150), (D.28) d'après (D.132), (D.29) d'après (D.131). Par conséquent, puisque (D.136) est vérifiée, en invoquant la Proposition D.1.3, pour $\eta > 1$ et $\bar{\Delta} = \Delta$, on peut montrer qu'il existe $\beta \in \mathcal{KL}$, $\sigma \in \mathcal{K}$, $\bar{\sigma} \in \mathcal{KK}$ telles que, pour tout $t \in [t_0, \infty)$:

$$ \max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max \left\{ \begin{array}{l} \beta(\max \{|\omega_1(x_1(t_0))|, |\omega_2^a(x_2(t_0))|\}, t - t_0), \\ \sigma(\max \{ \|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)} \}), \\ \bar{\sigma}(\tau^*, \max \{ \|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)} \}), \\ \eta \tilde{\sigma}^m(\tau^*, m), \eta m \end{array} \right\}. \quad (D.157) $$

Sachant que $\delta^m(\tau,s) = \max \{\eta\tilde{\sigma}^m(\tau,s), \eta s\}$, pour $s \in \mathbb{R}_{\ge 0}$, d'après (D.131):

$$ \max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max \left\{ \begin{array}{l} \beta(\max \{|\omega_1(x_1(t_0))|, |\omega_2^a(x_2(t_0))|\}, t - t_0), \\ \sigma(\max \{ \|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)} \}), \\ \bar{\sigma}(\tau^*, \max \{ \|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)} \}), \varepsilon \end{array} \right\}. \quad (D.158) $$

□

De la même manière, on peut déduire du Théorème D.2.2 le résultat suivant qui est utilisé au Chapitre 7. Il est intéressant de souligner que celui-ci peut être appliqué pour l'analyse de problèmes de poursuite de trajectoires ou de synchronisation lorsque les données du contrôleur sont échantillonnées en considérant le système comme étant paramétré en la période d'échantillonnage, cf. [163].

**Théorème D.3.2** *Considérons le système (D.124)-(D.125), supposons qu'il existe $\beta_1, \beta_2 \in \mathcal{KL}$, $\alpha_1, \alpha_2, \gamma_2^{\omega_1}, \gamma_2^{\omega_2}, \eta_2^{\omega_1}, \eta_2^{\omega_2}, \eta_2^{\omega_1} \in \mathcal{K}$, $\gamma_1^{\omega_2}, \gamma_1^{\omega_1}, \eta_1^{\omega_1}, \eta_1^{\omega_2}, \eta_1^{\omega_2}, \eta_1^{\omega_1} \in \mathcal{KK}$, $\bar{\tau} \in [\nu, \infty]$, tels que pour tout $x_1(t_0) \in \mathbb{R}^{n_{x_1}}$, $(u_1, x_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{x_2}}$, $\tau \in [\nu, \bar{\tau})$, et $t \ge t_0 \ge 0$, les solutions de (D.124)*
vérifient :

$$
\begin{equation}
\begin{aligned}
|\omega_1(x_1(t))| &\le \max \left\{ \beta_1(|\omega_1(x_1(t_0))|, t-t_0), \gamma_1^{\omega_2}(\tau, \|\omega_2^a(x_2)\|_{[t_0,t)}), \right. \\
&\qquad \left. \gamma_1^{\omega_2^b}(\tau, \|\omega_2^b(x_2)\|_{[t_0,t)}), \gamma_1^u(\tau, \|u_1\|_{[t_0,t)}) \right\}
\end{aligned}
\tag{D.159}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
|x_1(t)| &\le \max \left\{ \alpha_1(|x_1(t_0)|), \eta_1^{\omega_1}(\tau, \|\omega_1(x_1)\|_{[t_0,t)}), \eta_1^{\omega_2^a}(\tau, \|\omega_2^a(x_2)\|_{[t_0,t)}), \right. \\
&\qquad \left. \eta_1^{\omega_2^b}(\tau, \|\omega_2^b(x_2)\|_{[t_0,t)}), \eta_1^u(\tau, \|u_1\|_{[t_0,t)}) \right\},
\end{aligned}
\tag{D.160}
\end{equation}
$$

pour tout $x_2(t_0) \in \mathbb{R}^{n_{x_2}}$, $(x_1, u_2) \in \mathcal{L}_{\infty}^{n_{x_1}+n_{u_2}}$, $\tau \in [v, \bar{\tau})$, et $t \ge t_0 \ge 0$, les solutions de (D.125) vérifient :

$$
|\omega_2^a(x_2(t))| \le \max \left\{ \beta_2(|\omega_2(x_2(t_0))|, t-t_0), \gamma_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,t)}), \gamma_2^u(\|u_2\|_{[t_0,t)}) \right\} \quad (\text{D.161})
$$

$$
\begin{equation}
\begin{aligned}
|\omega_2^b(x_2(t))| &\le \max \left\{ \alpha_2(|\omega_2^b(x_2(t_0))|), \eta_2^{\omega_1}(\|\omega_1(x_1)\|_{[t_0,t)}), \eta_2^{\omega_2^a}(\|\omega_2^a(x_2)\|_{[t_0,t)}), \right. \\
&\qquad \left. \eta_2^u(\|u_2\|_{[t_0,t)}) \right\},
\end{aligned}
\tag{D.162}
\end{equation}
$$

et, pour tout $\Delta, \varepsilon \in \mathbb{R}_{>0}$, soit $M \in \mathbb{R}_{>0}$ (suffisamment grand) et $m \in \mathbb{R}_{>0}$ (suffisamment petit) tels que

$$
\left\{
\begin{array}{l}
\delta^m(\bar{\tau}, m) < \varepsilon \\
\max \{m, \bar{\sigma}^m(\bar{\tau}, m), \nu^x(\bar{\tau}, \Delta), \nu^u(\bar{\tau}, \Delta)\} < M,
\end{array}
\right.
\qquad (\text{D.163})
$$

où, pour $(\tau, s) \in [v, \bar{\tau}) \times \mathbb{R}_{\ge 0}$ :

$$
\delta^m(\tau, s) = \max \{ \eta \tilde{\sigma}^m(\tau, s), \eta s \}
$$

$$
\begin{equation}
\begin{aligned}
\nu^x(\tau,s) = \max & \left\{ \beta_1(\rho_1(s),0), \gamma_1^{\omega_2^a}(\tau, \beta_2(\bar{\rho}_2(s),0)), \gamma_1^{\omega_2^b}(\tau, \alpha_2(\bar{\rho}_2(s))), \right. \\
& \left. \gamma_1^{\omega_2^b}(\tau, \eta_2^{\omega_2^a}(\beta_2(\bar{\rho}_2(s),0))), \beta_2(\bar{\rho}_2(s),0), \gamma_2^{\omega_1}(\beta_1(\rho_1(s),0)), \right. \\
& \left. \gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\tau, \alpha_2(\bar{\rho}_2(s)))), \rho_1(s), \bar{\rho}_2(s), \tilde{\beta}(s,0), \tilde{\sigma}^x(\tau,s) \right\}
\end{aligned}
\tag{D.164}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\nu^u(\tau,s) = \max & \left\{ \gamma_1^{\omega_2^a}(\tau, \gamma_2^u(s)), \gamma_1^u(\tau, s), \gamma_1^{\omega_2^b}(\tau, \eta_2^{\omega_2^a}(\gamma_2^u(s))), \gamma_1^{\omega_2^b}(\tau, \eta_2^u(s)), \right. \\
& \left. \gamma_2^{\omega_1}(\gamma_1^u(\tau, s)), \gamma_2^u(s), \gamma_2^{\omega_1}(\gamma_1^{\omega_2^b}(\tau, \eta_2^u(s))), \tilde{\gamma}^u(\tau, s) \right\},
\end{aligned}
\tag{D.165}
\end{equation}
$$

avec $\eta > 1$, $\tilde{\sigma}^m \in KK$, $\tilde{\beta} \in KL$, $\tilde{\sigma}^x \in KK$ et $\tilde{\gamma}^u \in KK$ se déduisent respectivement de (D.118), (D.116), (D.119) et (D.122). Il existe $\tau^*(\varepsilon, \Delta) \in [v, \bar{\tau})$ (défini par (D.168)), que l'on note $\tau^*$, tel que

$$
\max \left\{ \gamma_1^{\omega_2^a}(\tau^*, \gamma_2^{\omega_1^b}(s)), \gamma_1^{\omega_2^b}(\tau^*, \eta_2^{\omega_2^a}(\gamma_2^{\omega_1^b}(s))), \gamma_1^{\omega_2^b}(\tau^*, \eta_2^{\omega_1^b}(s)), \gamma_2^{\omega_1^b}(\tau^*, \gamma_2^{\omega_2^a}(\gamma_2^{\omega_1^b}(s))), \right. \\
\left. \gamma_2^{\omega_1^b}(\gamma_1^{\omega_2^b}(\tau^*, \eta_2^{\omega_2^a}(s))), \gamma_2^{\omega_1^b}(\gamma_1^{\omega_2^b}(\tau^*, \eta_2^{\omega_1^b}(s))) \right\} < s \quad \text{pour } s \in [m, M], \tag{D.164}
$$

alors, il existe $\mu \in \mathbb{R}_{>0}$ tel que pour tout $\tau \in [v, \tau^*)$, $(x_1(t_0), x_2(t_0)) \in \mathbb{R}^{n_{x_1}+n_{x_2}}$ et $(u_1, u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ avec $\max \{|x_1(t_0)|, |x_2(t_0)|, \|u_1\|_{\infty}, \|u_2\|_{\infty}\} < \Delta$, les solutions de (D.124)-(D.125) vérifient :

$$
|(x_1(t), x_2(t))| \le \mu \quad \forall t \ge t_0 \ge 0. \tag{D.165}
$$

De plus, il existe $\beta \in KL$, $\sigma \in K$, $\bar{\sigma} \in KK$ tels que pour tout $\tau \in [v, \tau^*)$, $(x_1(t_0), x_2(t_0)) \in \mathbb{R}^{n_{x_1}+n_{x_2}}$ et $(u_1, u_2) \in \mathcal{L}_{\infty}^{n_{u_1}+n_{u_2}}$ avec $\max \{|x_1(t_0)|, |x_2(t_0)|, \|u_1\|_{\infty}, \|u_2\|_{\infty}\} < \Delta$, les solutions
de (D.124)-(D.125) satisfont, pour tout $t \ge t_0 \ge 0$,

$$
\begin{equation}
\begin{split}
\max \{|\omega_1(x_1(t))|, |\omega_2^a(x_2(t))|\} \le \max & \left\{ \beta(\max \{|\omega_1(x_1(t_0))|, |\omega_2(x_2(t_0))|\}, t-t_0), \right. \\
& \qquad \left. \sigma(\max \{\|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)}\}), \right. \\
& \qquad \left. \bar{\sigma}(\tau^*, \max \{\|u_1\|_{[t_0,t)}, \|u_2\|_{[t_0,t)}\}), \right. \\
& \qquad \left. \delta^x(\tau^*, \Delta), \varepsilon \right\},
\end{split}
\tag{D.166}
\end{equation}
$$

où $\delta^x(\tau,s) = \eta \tilde{\sigma}^x(\tau,s)$, pour $(\tau,s) \in [v, \bar{\tau}) \times \mathbb{R}_{\ge 0}$.

**Preuve abrégée.** De la même manière que la preuve du Théorème D.3.1 suit celle du Théorème D.2.1, on utilise ici le Théorème D.2.2 pour obtenir les résultats désirés. Il suffit pour cela de garantir l’existence d’un $\tau^* \in [v, \bar{\tau})$ tel que (D.164) est garantie. On définit pour cela la bijection suivante

$$
\begin{align}
\bar{\gamma} : [0, \tau_1) \times [0, s_1) &\longrightarrow [0, \tau_2) \times [0, s_2) \nonumber \\
(\tau, s) &\longmapsto \max \left\{ \begin{aligned}
& \gamma_1^{\omega_1}(\tau^*, \gamma_2^{\omega_1}(s)), \gamma_1^{\omega_2}(\tau^*, \eta_2^{\omega_1}(s)), \gamma_1^{\omega_2}(\tau^*, \eta_2^{\omega_2}(s)), \\
& \gamma_2^{\omega_1}(\tau^*, \gamma_1^{\omega_2}(s)), \gamma_2^{\omega_1}(\tau^*, \gamma_1^{\omega_2}(\tau^*, \eta_2^{\omega_2}(s))), \gamma_2^{\omega_1}(\tau^*, \gamma_1^{\omega_2}(\tau^*, \eta_2^{\omega_1}(s)))
\end{aligned} \right\} \tag{D.167}
\end{align}
$$

où $\tau_1, \tau_2, s_1, s_2 \in \mathbb{R}_{>0} \cup \{\infty\}$. Soit $\bar{\gamma}^{-1}$ l'inverse de $\gamma$ par rapport à $\tau$. On prend la convention que $\bar{\gamma}^{-1}(s) = \infty$ si $s \ge \tau_2$. En choisissant

$$
\tau^* = \min \left\{ \bar{\tau}, \min_{s \in [m, M]} \gamma^{-1}(s) \right\}, \qquad (D.168)
$$

(D.164) est bien satisfaite. La preuve se conclut en suivant celle du Théorème D.2.2 et en
prêtant attention aux dépendances de la même manière que pour le Théorème D.3.1. $\square$
# Annexe E

## Fonction de Lyapunov pour le protocole TOD modifié

Nous rappelons la définition du protocole TOD modifié présenté dans §4.3.2. La fonction $h$ est définie par, pour $e = (e_1, \dots, e_l) \in \mathbb{R}^{n_e} (l \in \mathbb{Z}_{>0})$:

$$h(e) = (\mathbb{I} - \Psi(e))e, \qquad (E.1)$$

et $\Psi(e) = \text{diag}(\Psi_1(e)\mathbb{I}_{n_1}, \dots, \Psi_l(e)\mathbb{I}_{n_l})$ avec, pour $i \in \{1, \dots, l\}$:

$$\Psi_i(e) = \begin{cases} \text{sat}(\sqrt{l}|e_i|) & \text{si } i = \min(\arg \max_{j \in \{1, \dots, l\}} |e_j|) \\ 0 & \text{sinon,} \end{cases} \qquad (E.2)$$

où $\text{sat}(s) = \min(s,1)$ pour tout $s \in \mathbb{R}_{\ge 0}$. Le protocole se comporte comme le protocole TOD pour de grands $|e|$, mais produit des sauts plus petit pour de faibles valeurs de $|e|$.

**Proposition E.0.1** *Le protocole TOD modifié défini par (E.1) est UGAS.*

*Preuve.* Soit $W(e) = |e|$ pour $e \in \mathbb{R}^{n_e}$ et supposons sans perte de généralité que $|e_1| \ge \max_{j \in \{2, \dots, l\}} |e_j|$ et $h_1(e) = (1 - \text{sat}(\sqrt{l}|e_1|))e_1$. On peut écrire que:

$$|e_1|^2 = \max_{j \in \{1, \dots, l\}} |e_j|^2 \ge \frac{1}{l} \sum_{j \in \{1, \dots, l\}} |e_j|^2 = \frac{1}{l} |e|^2. \qquad (E.3)$$

D'après (E.1),

$$\begin{align*}
|h(e)|^2 &= \sum_{j \in \{2, \dots, l\}} |h_j(e)|^2 + h_1(e)^2 \\
&= \sum_{j \in \{2, \dots, l\}} |e_j|^2 + (1 - \text{sat}(\sqrt{l}|e_1|))^2 |e_1|^2 \\
&= |e|^2 + \text{sat}(\sqrt{l}|e_1|) (\text{sat}(\sqrt{l}|e_1|) - 2) |e_1|^2, \tag{E.4}
\end{align*}$$
en appliquant (E.3) et remarquant que

$$ \mathrm{sat}(\sqrt{l}|e_1|) (\mathrm{sat}(\sqrt{l}|e_1|) - 2) \le 0 \qquad (E.5) $$

et que $s \mapsto |\mathrm{sat}(\sqrt{l}s)(\mathrm{sat}(\sqrt{l}s) - 2)|$ est croissante sur $\mathbb{R}_{\ge 0}$ (puisque $\mathrm{sat}$ est croissante sur $\mathbb{R}_{\ge 0}$), on a :

$$ \begin{aligned} |h(e)|^2 &= |e|^2 - |\mathrm{sat}(\sqrt{l}|e_1|)(\mathrm{sat}(\sqrt{l}|e_1|) - 2)| |e_1|^2 \\ &\le |e|^2 - \frac{1}{l} |\mathrm{sat}(|e|)(\mathrm{sat}(|e|) - 2)| |e|^2 \\ &= \left[ 1 + \frac{1}{l} \mathrm{sat}(|e|)(\mathrm{sat}(|e|) - 2) \right] |e|^2. \end{aligned} \qquad (E.6) $$

Ainsi, pour tout $e \in \mathbb{R}^{n_e}$,

$$ \begin{aligned} W(h(e)) - W(e) &= |h(e)| - |e| \\ &\le -\alpha_3(W(e)), \end{aligned} \qquad (E.7) $$

où $\alpha_3(s) = [1 - \sqrt{1 + \frac{1}{l}\mathrm{sat}(s)(\mathrm{sat}(s) - 2)}]$ s avec $\alpha_3 \in \mathcal{K}$. Pour $W(e) \ge 1$, on retrouve la même inégalité que pour le TOD :

$$ W(h(e)) \le \sqrt{\frac{l-1}{l}} W(e). \qquad (E.8) $$

Considérons le cas où $W(e) \le 1$. Une fonction $\varrho \in \mathcal{K}_\infty$ est construite, comme dans la preuve de la Proposition 4.3.1, de telle sorte que, pour $\hat{W} = \varrho \circ W$ et tout $e \in \mathbb{R}^{n_e}$,

$$ \hat{W}(h(e)) \le \sqrt{\frac{l-1}{l}} \hat{W}(e), \qquad (E.9) $$

on prend par exemple :

$$ \varrho(s) = \begin{cases} \exp\left(\int_1^s \frac{2}{a(\eta)} d\eta\right) & \text{pour } s > 0 \\ 0 & \text{pour } s = 0, \end{cases} \qquad (E.10) $$

avec $a(s) = \frac{2}{\pi} \int_0^s \frac{\min\{\eta, \frac{2\sqrt{l}}{\sqrt{l-1}\alpha_3(\eta)}\}}{1+\eta^2} d\eta$. On remarque que, d'après la preuve de la Proposition 4.3.1, les propriétés Lipschitziennes de $W$ sont également vérifiées par $\hat{W}$ puisque $\rho \in C^1(\mathbb{R}_{\ge 0}, \mathbb{R}_{\ge 0})$. Soit

$$ \tilde{W}(e) = \begin{cases} W(e) & \text{pour } W(e) \ge 1 \\ \hat{W}(e) & \text{pour } W(e) < 1, \end{cases} \qquad (E.11) $$

$\tilde{W}$ est continue sur $\mathbb{R}^{n_e}$ puisque $\varrho(1) = 1$. Le protocole TOD modifié est bien UGAS, d'après la Définition 4.3.1 avec

$$ \alpha_1(s) = \alpha_2(s) = \begin{cases} s & \text{pour } s \ge 1 \\ \varrho(s) & \text{pour } s \in [0,1), \end{cases} \qquad (E.12) $$

qui sont bien de classe $\mathcal{K}_\infty$ et $\rho = \sqrt{\frac{l-1}{l}}$. De plus on peut montrer que $\tilde{W}$ est localement Lipschitzienne, sachant que $W$ et $\hat{W}$ le sont et que $\tilde{W}$ est continue sur $\mathbb{R}_{\ge 0}$. $\square$
# Annexe F

Observateur adaptatif
continu-discret pour des systèmes
affines en l'état
# Continuous-discrete adaptive observers for state affine systems

Tarek Ahmed-Ali<sup>a</sup>, Romain Postoyan<sup>b</sup>, Françoise Lamnabhi-Lagarrigue<sup>c</sup>

## Abstract

The observation of a class of multi-input multi-output (MIMO) state affine systems with constant unknown parameters and discrete time output measurements is addressed. Assuming some persistent excitation conditions to hold and the sampling steps to satisfy some boundedness hypotheses, system observability is ensured and a class of global exponential observers is synthesized.

**Key words:** Sampled-data observer, adaptive observer.

## 1 Introduction

The synthesis of observers for continuous-time systems with sampled measurements has received much attention over the past decades. A classical solution used to handle the discrete nature of the system output consists either in appropriately discretizing a known continuous time observer [2] or in synthesizing a discrete-time observer for a consistent approximation of the exact discretized system [11,2] for instance. Numerical schemes can also be applied [20,7], but such observers usually require high computational capacities [7] and are only local. An alternative to these approaches is to design a continuous-time observer fed by discrete-time system measures and thus called *continuous-discrete* observer. This approach is particularly interesting when the observer dynamics is much faster than the system one (that is the case when high speed processor computers are used). Therefore, the approximation of having a continuous-time observer, although implemented digitally, is acceptable. Moreover, contrary to the discrete-time approach, obtained stability properties are global and asymptotic (in most cases) [12,13,22] and not semiglobal practical like in the discrete-time formulation [2].

Sampled system measures imply some additional difficulties for the observation compared with the case where outputs are continuously known. Indeed, despite the fact that continuous observation methods have to be adapted to the periodic availability of the output, the system observability also depends on the sampling times. For obvious reasons, if the time interval between two successive measures is too large, the system may lose its observability, as shown in [22]. Furthermore, considering nonlinear systems with inputs, it is well known that there may exist inputs which make the system unobservable. Hence, the notion of *universal input* has been introduced to denote the class of inputs which make considered systems observable. This class of inputs, called *regularly persistent* inputs, has been characterized for some classes of state affine systems in [8,9,14].

Existing continuous-discrete time observers are mostly based on extended Kalman filter techniques. In [12], a class of global exponential observers has been synthesized for multi-input single-output nonlinear systems put into a canonical form. The principle is to divide the observation into two steps: one, called *prediction*, between sampling times, that consists of ‘copying’ system dynamics with no correction term and updating the dynamical observation gain, and another one, at the sampling times, where the error between the system and the observer output is used to correct the estimate state trajectory. Deriving some conditions on the sampling times through the observer stability analysis, the exponential convergence of the observation error is then

<footnote><sup>a</sup>Univ Caen, GREYC-ENSICAEN, UMR CNRS 6072-6 Boulevard du Maréchal Juin 14050 Caen Cedex</footnote>

<footnote><sup>b</sup>Univ Paris-Sud, LSS-CNRS Supélec, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France</footnote>

<footnote><sup>c</sup>EECI, LSS-CNRS Supélec, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France</footnote>

<footnote>* Corresponding author T. Ahmed-Ali. Tel. +33(0)298348856. Fax +33(0)298348746</footnote>

<footnote>Email addresses: tarek.ahmed-ali@greyc.ensicaen.fr (Tarek Ahmed-Ali), postoyan@lss.supelec.fr (Romain Postoyan), lamnabhi@lss.supelec.fr (Françoise Lamnabhi-Lagarrigue).</footnote>
ensured. The work of [12] has been extended to other classes of systems. In [21], observers for a MIMO class of state affine systems where the dynamical matrix depends on the input, have been designed when the inputs are regularly persistent. In [3], a similar method has been used for a larger class of systems and applied to the observation of emulsion copolymerization process. The observation of a class of systems with output injection has been treated in [22]. Recently, in [15], a high gain continuous-discrete observer has been developed but using constant observation gains.

On the other hand, it appears that for a number of physical processes some parameters may not be known, that makes existing results on continuous-discrete time observers non-applicable. In this paper, the observation of the class of systems studied in [21] is extended to the adaptive case. Early developments of adaptive observers were made in [16,17] for linear systems. Adaptive observation of nonlinear systems have been investigated using different techniques that basically rely on linear adaptive algorithms, through coordinate change or output injection for instance, see [4,5,10,19,23]. In [24], a unified interpretation of the latter references has been proposed that emphasizes their characteristics. In [6], adaptive observation for state affine systems in continuous-time is discussed and the work [23] is extended. In this study, the approach developed in [6] on extended Kalman filters for a class of MIMO linear time-variant, is adapted to the sampled measures problem at the difference that, here, the estimation law is based on discrete-time adaptive techniques.

After having defined the class of systems considered and recalled the main objective, a class of adaptive continuous-discrete observers is designed. Assuming some persistent excitation conditions and some hypotheses on the system output sampling times to hold, the global exponential stability of the observation error is proved. Finally, a simulation example is performed which illustrates the design procedure.

## 2 Nomenclature

First some mathematical notation is introduced. Let $\mathbb{R}^\text{def} = (-\infty, \infty)$, $\mathbb{R}_+^\text{def} = (0, \infty)$, $\mathbb{R}_0^+ = [0, \infty)$ and defined the Euclidean norm $\|\cdot\|$. For $p, q, n, m \in \mathbb{N}$, $\mathbb{R}^{p \times q}$ represents the set of real matrices of order $p \times q$ and $\mathbb{I}_p \in \mathbb{R}^{p \times p}$ stands for the identity matrix of order $p \times p$. If $\mathcal{X} \subset \mathbb{R}^{p \times q}$ and $\mathcal{Y} \subset \mathbb{R}^{n \times m}$, $\mathcal{C}(\mathcal{X}, \mathcal{Y})$ denotes the space of all continuous functions mapping $\mathcal{X} \to \mathcal{Y}$. If $P \in \mathbb{R}^{p \times p}$, $P > 0$ means that $P$ is positive definite. The notation $\|P\|$, for $P \in \mathbb{R}^{p \times q}$, represents the $L_2$-norm of $P$. For $A: \mathbb{R} \to \mathbb{R}^{p \times q}$ and $t \in \mathbb{R}$, the notation $A(t^{-})$ denotes the left limit of $A$ at instant $t$, if it exists. In all this study, the initial time is called $t_0 \in \mathbb{R}$.

## 3 Problem statement

The following class of systems is considered, for $t \in [t_k, t_{k+1})$, $k \ge 0$,

$$ 
\begin{aligned}
\dot{x}(t) &= A(u)x(t) + b(u) + \phi(u)\theta, \\
y(t_k) &= Cx(t_k),
\end{aligned}
\qquad (1) $$

where $x \in \mathbb{R}^n$ is the instantaneous state vector, $u \in \mathcal{D} \subset \mathbb{R}^m$ the input vector ($\mathcal{D}$ is compact), $y \in \mathbb{R}^p$ is the output vector, $\theta \in \mathbb{R}^l$ is a vector of unknown constant parameters, $A \in \mathcal{C}(\mathbb{R}^m, \mathbb{R}^{n \times n})$, $b \in \mathcal{C}(\mathbb{R}^m, \mathbb{R}^n)$, $C \in \mathbb{R}^{p \times n}$, $\phi \in \mathcal{C}(\mathbb{R}^m, \mathbb{R}^{n \times l})$ are known, with $n, m, p, l \in \mathbb{N}$ and $x_0 = x(t_0)$. The notation $(t_k)_{k \ge 0}$ represents a strictly increasing sequence such that $\lim_{k \to \infty} t_k = \infty$ that models the sampling times. The maximum sampling step is denoted $\tau = \max_{k \ge 0}(\tau_k)$, where $\tau_k = t_{k+1} - t_k$.

Note that the class of systems (1) contains systems of the form, for $k \ge 0$,

$$ 
\begin{aligned}
\dot{x}(t) &= A(t, u)x(t) + b(t, u) + \phi(t, u)\theta, \quad t \in [t_k, t_{k+1}), \\
y(t_k) &= Cx(t_k), \quad t = t_{k+1},
\end{aligned}
\qquad (2) $$

since the dependence on $u$ can be considered like a time dependence.

The main objective of this paper is to synthesize a global exponential adaptive observer, as defined in Section 4.3, for system (1).

## 4 Observer design and stability study

### 4.1 Observation structure

The proposed observer can be viewed as an extension to the adaptive case of the structure developed in [21]. An auxiliary variable, $\lambda$, which plays a key role in the convergence of parameters estimator, is notably introduced like in [23,6].

For $k \ge 0$, $t \in [t_k, t_{k+1})$,

$$ \dot{\hat{x}}(t) = A(u)\hat{x}(t) + b(u) + \phi(u)\hat{\theta}(t_k), \qquad (3a) $$

$$ \dot{S}(t) = -A(u)^{\mathrm{T}}S(t) - S(t)A(u) - \mu S(t), \qquad (3b) $$

$$ \dot{\lambda}(t) = A(u)\lambda(t) + \phi(u), \qquad (3c) $$

$$ \dot{\theta}(t) = 0, \qquad (3d) $$

for $t = t_{k+1}$,

$$ 
\begin{aligned}
\hat{x}(t_{k+1}) = \hat{x}(t_{k+1}^{-}) &+ (\lambda(t_{k+1})\Delta(t_{k+1}) \\
&+ \rho\tau_k S^{-1}(t_{k+1})C^{\mathrm{T}})(y(t_{k+1}) - C\hat{x}(t_{k+1}^{-})),
\end{aligned}
\qquad (4a) $$
$$S(t_{k+1}) = S(t_{k+1}^{-}) + \tau_k C^T C, \quad (4b)$$

$$\lambda(t_{k+1}) = M(\tau_k, t_{k+1})\lambda(t_{k+1}^{-}), \quad (4c)$$

$$\hat{\theta}(t_{k+1}) = \hat{\theta}(t_k) + \Delta(t_{k+1})(y(t_{k+1}) - C\hat{x}(t_{k+1})), \quad (4d)$$

with, $\Delta(t_{k+1}) = \frac{\epsilon}{1+\epsilon}\|\lambda(t_{k+1}^{-})^T C^T \Sigma(t_{k+1}^{-}) C \lambda(t_{k+1}^{-})\| \lambda(t_{k+1}^{-})^T C^T \Sigma(t_{k+1}^{-})$, and $M(\tau_k, t_{k+1}) = (\mathbb{I}_n - \rho\tau_k S^{-1}(t_{k+1})C^T C)$.

The notation $\hat{x} \in \mathbb{R}^n$ and $\hat{\theta} \in \mathbb{R}^l$ respectively denote the estimate of the state, $x$, and the estimate of the unknown vector parameter $\theta$, $S(t) \in \mathbb{R}^{n \times n}$ is the so-called observation gain, $\lambda \in \mathbb{R}^n$, $\Sigma \in \mathcal{C}(\mathbb{R}, \mathbb{R}^{p \times p})$ is symmetric positive definite for all $t$, and $\mu, \epsilon \in \mathbb{R}_+$, $\rho \in [1, \infty)$ are some design parameters. Note that the existence of the inverse of matrix $S$ will be ensured in the following. The initial conditions are denoted $\hat{x}_0 \in \mathbb{R}^n$, $S_0 > 0$ symmetric, $\lambda_0 \in \mathbb{R}^n$ and $\hat{\theta}_0 \in \mathbb{R}^l$.

The observer (3a-d)-(4a-d) is composed of a predictor part (3a-d) and a correction one (4a-d). During $[t_k, t_{k+1})$, the state estimate $\hat{x}$ has the same dynamics than system (1) but using the estimate of the unknown variable, $\hat{\theta}$, that is kept constant (3d). As commonly done for high gain observers, the matrix $S$ is defined as the solution of the differential equation (3b), except that a term of the form $C^T C$ is usually added in order to ensure $S$ to be positive definite, see for example in [21]. At each sampling instants, observer trajectory is corrected taking into account the available measure (4a). Thank to (4b), conditions on the inputs and restriction on $\tau$, it will be shown that matrix $S$ remains positive definite. The dynamics of $\hat{\theta}$ are defined by a least mean square law (4d), that will guarantee the convergence of the estimate to the true value, under some persistent excitation conditions. The usefulness of variable $\lambda$ will be shown in the stability analysis, like in [23,6].

**Remark 1** If the parameter vector $\theta$ is equal to zero, system (3a-d)-(4a-d) has the same form than in [21].

The dynamical equation satisfied by the state observation error, $e = x - \hat{x}$, is, in view of (1), (3a-d) and (4a-d), for $k \ge 0$:

$$\dot{e}(t) = A(u)e(t) + \phi(u)\tilde{\theta}(t_k), \quad t \in [t_k, t_{k+1}), \quad (5a)$$

$$e(t_{k+1}) = (M(\tau_k, t_{k+1}) - \lambda(t_{k+1})\Delta(t_{k+1}C))e(t_{k+1}^{-}), \quad t = t_{k+1}. \quad (5b)$$

The following persistent excitation condition is assumed to hold for system (1) in order to guarantee the identifiability of vector $\theta$ [23,6].

**Hypothesis 2** Let $\lambda(\cdot) \in \mathbb{R}^{n \times s}$ be the matrix solution of the impulsive ordinary differential equation (3c)-(4c). The matrix $\lambda(\cdot)$ is persistently excited so that there exist $k_1 \in \mathbb{N}$, $\delta > 0$ and $\Sigma(\cdot) \in \mathcal{C}([t_0, \infty), \mathbb{R}^{p \times p})$ a time-dependent bounded symmetric positive definite matrix,

such that, for all $k \ge 0$,

$$\sum_{j=k}^{k+k_1} \lambda^T(t_j^-) C^T \Sigma(t_j) C \lambda(t_j^-) \ge \delta \mathbb{I}_n. \quad (6)$$

Before stating the main theorem, some preliminary definition and results are required.

**Definition 3** Consider system, for $t \in [t_0, \infty)$:

$$\begin{aligned} \dot{x}(t) &= A(u)x(t), \\ y(t) &= Cx(t), \end{aligned} \quad (7)$$

and the transition matrix $\Psi_u(\cdot, t_0)$ associated to system (7), such that $\Psi_u(t_0, t_0) = \mathbb{I}_n$, the bounded input $u$ is said to be regularly persistent if there exist $t_1, t_2 \in [t_0, \infty)$, $\alpha \in \mathbb{R}_+$, such that, for all $t \ge t_2$,

$$\int_t^{t+t_1} \Psi_u^T(s, t_0) C^T C \Psi_u(s, t_0) \, ds \ge \alpha \mathbb{I}_n. \quad (8)$$

As mentioned in Introduction, regularly persistent inputs are the class of inputs that guarantee the system to be observable. For more details, see [8,9,14].

**Remark 4** In practice, condition (8) can be checked off-line, when $u(t)$ is known for all $t \in [t_0, \infty)$, using numerical tools for approximating the integral term. Thus, coefficients $t_1$ and $\alpha$ are obtained off-line in order to derive a sampling period that satisfies the bound given in (9).

## 4.2 Technical results

The following proposition guarantees that matrix $S$ is positive definite for any $t \in [t_0, \infty)$, under conditions on the types of inputs applied to the system and $\tau$.

**Proposition 5** Let $u$ be a regularly persistent input for system (1), for all $\mu \ge \xi$ where $\xi = 2 \sup_{t \ge t_0} \|A(u(t))\|$, if $\tau \le \bar{\tau}$, where $\bar{\tau}$ is the unique positive term such that $\bar{\tau} = \frac{\alpha e^{-\xi(t_1+\bar{\tau})}}{2\xi \|C^T C\|(t_1+\bar{\tau})}$, then, for all $S(t_0)$ symmetric positive definite, there exist constants $\beta_1, \beta_2 \in \mathbb{R}_+$ such that, for $t \in [t_0, \infty)$,

$$\beta_1 \mathbb{I}_n \le S(t) \le \beta_2 \mathbb{I}_n.$$

The proof of Proposition 5 can be found in [21].

**Remark 6** a) Since considered inputs belong to a compact set, $\mathcal{D}$, and because $A$ is continuous, $\xi$ is well defined. b) There always exists such a $\bar{\tau}$; indeed writing condition $\bar{\tau} = \frac{\alpha e^{-\xi(t_1+\bar{\tau})}}{2\xi \|C^T C\|(t_1+\bar{\tau})}$ as $2\xi \|C^T C\|(t_1+\bar{\tau})\bar{\tau} = \alpha e^{-\xi(t_1+\bar{\tau})}$, loosely speaking it is clear that, for positive values of $\bar{\tau}$,
the polynomial term on the left-hand side will 'cross' the exponential on the right-hand side at a unique point.

c) Considering regularly persistent inputs, the positive definition of the solution S of equations (3b)-(4b) can only be ensured under condition on the sampling steps, otherwise the solution of equation (3b) may become non-positive definite [21].

d) Notice that $\beta_1, \beta_2$ implicitly depend on the maximum allowable sampling interval $\bar{\tau}$.

**Lemma 7** Suppose the following system to be globally exponentially stable,

$$ \zeta(t_{k+1}) = F(t_k)\zeta(t_k), $$

and let u be a bounded and integrable function of time that vanishes exponentially fast, then, system

$$ z(t_{k+1}) = F(t_k)z(t_k) + u(t_k), \quad k \ge 0, $$

converges exponentially to 0.

**Remark 8** This lemma is a straight extension of Lemma III.1, in [18], for discrete-time systems.

## 4.3 Stability analysis

System (3a-d)-(4a-d) is said to be a global exponential adaptive observer for system (1) if, along solutions to (1), (3a-d)-(4a-d):

(i) for all $(x_0, \hat{x}_0, \lambda_0, \hat{\theta}_0) \in \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^l$ and any $S_0 > 0$ symmetric, there exist $a_1, a_2, b_1, b_2 \in \mathbb{R}_+$, such that for all $t \ge t_0$, $\|e(t)\| \le a_1 e^{-b_1(t-t_0)} \|e_0\|$ and $\|\tilde{\theta}(t)\| \le a_2 e^{-b_2(t-t_0)} \|\tilde{\theta}_0\|$.

(ii) there exist $\bar{\lambda}, \bar{S} \in \mathbb{R}_+$, such that, for all $t \ge t_0$, $\|\bar{\lambda}(t)\| \le \bar{\lambda}, \|\bar{S}(t)\| \le \bar{S}$.

Condition (i) ensure the exponential convergence of the state and variable estimates to the corresponding true value, whereas (ii) ensure that the other observer variables do not explode in (in)finite time.

**Theorem 9** Assuming Hypothesis 2 to hold, the input to be regularly persistent, and choosing $\mu \ge \xi$ if,

$$ \tau \le \min\left\{\frac{(2\rho-1)\beta_1}{\rho^2\|C\|^2}, \bar{\tau}\right\}, \quad (9) $$

then system (3a-d)-(4a-d) is a global exponential adaptive observer for system (1).

**Proof.** Like in [23,6], the variable $\eta$ is introduced as $\eta = e - \lambda \tilde{\theta}$, where $\tilde{\theta} = \theta - \hat{\theta}$. Firstly, the global exponential convergence of $\eta$ to zero, along the solutions to dynamical equations to (1) and (3a-d)-(4a-d), is

proved. After having noticed that variables $\lambda$ and $S$ remain bounded, invoking Lemma 7, the exponential convergence to the origin of errors $e$ and $\tilde{\theta}$ is deduced.

The candidate Lyapunov function is defined as, for $t \ge t_0$:

$$ V(t) = \eta(t)^{\mathrm{T}}S(t)\eta(t). $$

Note that, according to Proposition 5 and since (9) is satisfied, the symmetric matrix $S$ is positive definite and of bounded norm. Let $k \ge 0$.

1. Let $t \in [t_k, t_{k+1})$. The variable $\eta$ satisfies the following dynamical equation, in view of (5a) and because, here, $\dot{\tilde{\theta}} = 0$ (3d):

$$ \begin{aligned} \dot{\eta}(t) &= A(u)e(t) + \phi(u)\tilde{\theta}(t_k) - \dot{\lambda}(t)\tilde{\theta}(t_k) \\ &= A(u)\eta(t) + (A(u)\lambda(t) + \phi(u) - \dot{\lambda}(t))\tilde{\theta}(t_k). \end{aligned} $$

Thus, according to (3c),

$$ \dot{\eta}(t) = A(u)\eta(t). \quad (10) $$

Consequently, differentiating the Lyapunov function along the solutions of (10),

$$ \dot{V}(t) = \eta(t)^{\mathrm{T}} (S(t)A(u) + A(u)^{\mathrm{T}}S(t) + \dot{S}(t)) \eta(t). $$

From (3b),

$$ \dot{V}(t) = -\mu V(t). \quad (11) $$

Integrating (11) over $[t_k, t]$,

$$ V(t) = e^{-\mu(t-t_k)}V(t_k), \quad (12) $$

and so,

$$ V(t_{k+1}^{-}) = e^{-\mu \tau_k} V(t_k). \quad (13) $$

2. Consider now $t = t_{k+1}$, it can be shown that, in view of (5b),

$$ \begin{aligned} \eta(t_{k+1}) &= (M(\tau_k, t_{k+1}) - \lambda(t_{k+1})\Delta(t_{k+1})C)e(t_{k+1}^{-}) \\ &\quad -\lambda(t_{k+1})\tilde{\theta}(t_{k+1}) \\ &= M(\tau_k, t_{k+1}) (\eta(t_{k+1}^{-}) + \lambda(t_{k+1}^{-})\tilde{\theta}(t_{k+1}^{-})) \\ &\quad -\lambda(t_{k+1}) (\Delta(t_{k+1})Ce(t_{k+1}^{-}) + \tilde{\theta}(t_{k+1}^{-})). \end{aligned} $$

Remarking that $\tilde{\theta}(t_{k+1}^{-}) = \tilde{\theta}(t_k)$ in view of (3d), and using (4b-d),

$$ \eta(t_{k+1}) = M(\tau_k, t_{k+1})\eta(t_{k+1}^{-}). \quad (14) $$
According to (4d),

$$
\begin{align*}
V(t_{k+1}) &= \eta(t_{k+1})^\mathrm{T} S(t_{k+1}) \eta(t_{k+1}) \\
&= \eta(t_{k+1}^-)^\mathrm{T} M(\tau_k, t_{k+1})^\mathrm{T} \\
&\quad \times (S(t_{k+1}^-) + \tau_k C^\mathrm{T} C) M(\tau_k, t_{k+1}) \eta(t_{k+1}^-) \\
&= V(t_{k+1}^-) - (2\rho - 1)\tau_k \|C\eta(t_{k+1}^-)\|^2 \\
&\quad + \rho^2 \tau_k^2 \eta(t_{k+1}^-) C^\mathrm{T} C S^{-1}(t_{k+1}) C^\mathrm{T} C \eta(t_{k+1}^-).
\end{align*}
$$

Invoking Proposition 5, the following inequality holds:

$$
V(t_{k+1}) \le V(t_{k+1}^-) - \tau_k \left( (2\rho - 1) - \frac{\rho^2 \tau_k \|C\|^2}{\beta_1} \right) \times \|C\eta(t_{k+1}^-)\|^2. \quad (15)
$$

Using (13) and because (9) holds,

$$
V(t_{k+1}) \le e^{-\mu \tau_k} V(t_k). \quad (16)
$$

Consequently, according to (12) and (16), for all $t \ge t_0$,

$$
V(t) \le e^{-\mu(t-t_0)} V(t_0), \\
\|\eta(t)\| \le \sqrt{\frac{\beta_2}{\beta_1}} e^{-\frac{\mu}{2}(t-t_0)} \|\eta(t_0)\|,
$$

the exponential stability of η = 0 for system (10)-(14) is
proved. Using similar arguments than for variable η, in
view of (3c) and (4c) and since function φ is continuous
and u takes values in a compact set, it can be easily
shown that there exists ¯ λ ∈ R + such that || λ ( t ) || ≤ ¯ λ,
for t ≥ t 0 . On the other hand, according to (4d),

$$
\tilde{\theta}(t_{k+1}) = (\mathbb{I}_n - \Delta(t_{k+1})C\lambda(t_{k+1}^-)) \tilde{\theta}(t_k) \\
- \Delta(t_{k+1})C\eta(t_{k+1}^-). \tag{17}
$$

Attention is focused on the homogenous part of (17),
which is typical of the normalized least mean square
algorithm:

$$
\tilde{\theta}(t_{k+1}) = (\mathbb{I}_n - \Delta(t_{k+1})C\lambda(t_{k+1}^-)) \tilde{\theta}(t_k). \quad (18)
$$

Using Theorem 2.8 in [1], since Hypothesis 2 holds, ˜ θ = 0 is globally exponentially stable for system (18). The non-homogenous part of equation (17) vanishes exponentially fast, because of the exponential convergence of η to 0 and the boundedness of Δ. Therefore, applying Lemma 7, ˜ θ = 0 is globally exponentially stable for system (17). Writing the observation error as e = η + λ ˜ θ , the exponential convergence of e to 0, along the solutions of (5a-b), can be deduced from the facts that ˜ θ and η converge with an exponential rate to zero and that λ is bounded. □

**Remark 10** Throughout the paper, no stability assumption is supposed to hold for A. In view of Theorem 9, even unstable matrices can be considered.

**Remark 11** The sampling interval must satisfy (9). In practice, after having numerically found admissible t₁ and α, the upper bound in (9) can be determined using parameters values.

4.4 Discussion on parameters choice

The selection of the observer parameters µ, ρ, ϵ illus-
trate the logical compromise between convergence speed
and bound on τ.

As it can be seen through (3b), parameter µ (taken big-
ger that ξ in view of Theorem 9) acts on the observation
gain in such a way that taking big values will increase
the convergence speed of the observer at the price of a
smaller admissible τ. Indeed, this will make the solution
of (3b) goes faster to non-positive definite matrices and
then decrease the value of β₁ (see Remark 6d). Concern-
ing ρ, a quick analysis could conclude that ρ = 1 is the
optimal value to be chosen in view of (9), but a bigger
value will help η to converge faster to 0, again, under
more severe conditions on τ (see (14) and the definition
of M). On the other hand, parameter ε allows to mod-
ulate the convergence of the parameter estimate.

5 Illustrative example

For illustrative purposes, consider a second-order system
modeled by, for $k \ge 0$ and $t \in [t_k, t_{k+1})$,

$$
\begin{align}
\dot{x}_1 &= \sin(u)x_2, \notag \\
\dot{x}_2 &= -0.1 \sin(u)x_1 - 0.2x_2 + 0.1 \cos(u)\theta, \tag{19} \\
y &= x_1, \notag
\end{align}
$$

where θ is unknown and u(t) = 2 - (cos(0.1t))². The
sampling step is chosen constant at τ = 0.11. An ob-
server of the form (3a-d)-(4a-d) is designed for system
(19). The design parameters have been fixed at ρ =
1, ϵ = 1, μ = 2.1 and Σ = 1. Using numerical tools,
the following values have been found α = 0.5, t₁ = 0.01,
ξ = 2.0396, β₁ ≥ 0.1125, consequently, condition (9) is
satisfied: τ ≤ min{0.1125, 0.21}. Therefore, since the
input is a persistent input for system (19), Theorem
9 applies. The initial conditions are t₀ = 0, x₁(0) =
x₂(0) = 50, ˆx₁(0) = ˆx₂(0) = ˆθ(0) = 0 and θ = 2.
For comparative purpose, the Euler discretization of ob-
server (11-15) in [6] has been simulated. Figure 1 shows
that, contrary to the continuous-time observer, the state
ˆx₂ given by the emulation does not converge asymptot-
ically to x₂ but oscillates. As mentioned in the Intro-
duction, this fact is not surprising since only semiglobal
practical stability properties can be established when
using the emulation of a continuous-time observer [2]. Moreover, the behaviour of emulation states present a big overshoot by opposite to the results obtained by the observer (3a-d)-(4a-d). Obtained parameter estimates converge slowly asymptotically to the true value for the continuous-discrete observer, and again, practically and with large overshoots for the emulation. Although not presented here, simulation results show that the continuous-time observer still works efficiently for larger values of τ like τ = 3 whereas, for such a value, the emulation states explode.

<figure><img src="image_37.png" /><figcaption>Figure 1. State $x_1$ and convergence of the state observation errors</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>Figure 2. Convergence of the parameter estimate</figcaption></figure>

# 6 Conclusions

The adaptive observation of a class of continuous MIMO systems with sampled measurements has been realized. Assuming the input acting on the system to satisfy some persistent excitation conditions and the sampling steps to respect given bounds, a class of global exponential observers has been developed.

# References

[1] B.D.O. Anderson, R.R. Bitmead, C.R. Johnson, P.V. Kokotović, R.L. Kosut, I.M.Y. Mareels, L. Praly, and

B.D. Riedle. *Stability of Adaptive Systems: Passivity and Averaging Analysis*. Signal Processing, Optimization and Control, MIT Press, Cambridge, U.S.A., 1986.

[2] M. Arcak and D. Nešić. A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation. *Automatica*, 40:1931–1938, 2004.

[3] C.-M. Astorga, N. Othman, S. Othman, H. Hammouri, and T.-F. McKenna. Nonlinear continuous-discrete observers: applications to emulsion polymerization reactors. *Control Eng. Practice*, 10:3–13, 2002.

[4] G. Bastin and M. Gevers. Stable adaptive observers for nonlinear time varying systems. *IEEE Trans. on Aut. Control*, 33(7):650–658, 1988.

[5] G. Besançon. Remarks on nonlinear adaptive observer design. *Syst. & Control Lett.*, 41(7):271–280, 2000.

[6] G. Besançon, J. De León-Morales, and O. Huerta-Guevara. On adaptive observers for state affine systems. *Int. J. of Contr.*, 79(6):581–591, 2006.

[7] E. Bıyık and M. Arcak. A hybrid redesign of newton observers in the absence of an exact discrete-time model. *Systems & Control Letters*, 55:429–436, 2006.

[8] G. Bornard, F. Celle, and N. Couenne. Regularly persistent observers for bilinear systems. In *Proc. of the 29th International Conference on Nonlinear Systems, New Trends in Nonlinear Systems Theory*, volume 122, pages 130–140, Nantes, France, 1988.

[9] F. Celle, J.P. Gauthier, D. Kazakos, and G. Sallet. Synthesis of nonlinear observers: a harmonic-analysis approach. *Math. Syst. Theory*, 22:291–322, 1989.

[10] Y. M. Cho and R. Rajamani. A systematic approach to adaptive observer synthesis for nonlinear systems. *IEEE Trans. on Aut. Control*, 42:534–537, 1997.

[11] A.M. Dabroom and H.K. Khalil. Output feedback sampled-data control of nonlinear systems using high-gain observers. *IEEE Trans. on Aut. Control*, 46(11):1712–1725, 2001.

[12] F. Deza, E. Busvelle, J.P. Gauthier, and D. Rakotopora. High gain estimation for nonlinear systems. *Sys. & Control Lett.*, 18:295–299, 1992.

[13] H. Hammouri, P. Kabore, S. Othman, and J. Biston. Failure diagnosis and nonlinear observer. application to a hydraulic process. *J. of the Franklin Institute*, 339:455–478, 2002.

[14] H. Hammouri and J. DeLeon Morales. Chapter: Topological properties of observer's inputs. In *Progress in Syst. and Control Theory, Birkhauser*, Boston, U.S.A., 2002.

[15] H. Hammouri, M. Nadri, and R. Mota. Constant gain observer for continuous-discrete time uniformly observable systems. In *Proc. of the 45th IEEE Conf. on Dec. & Control*, pages 6240–6244, San Diego, U.S.A., 2006.

[16] G. Kresselmeier. Adaptive observers with exponential rate of convergence. *IEEE Trans. on Aut. Control*, AC-22:2–8, 1977.

[17] G. Lüders and K.S. Narendra. An adaptive observer and identifier for a linear system. *IEEE Trans. on Aut. Control*, AC-18:496–499, 1973.

[18] R. Marino and P. Tomei. Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems. *IEEE Trans. on Aut. Control*, 40:1300–1304, 1995.

[19] R. Marino and P. Tomei. *Nonlinear control design*. ser. Information and system sciences, Prentice Hall, London, U.K., 1995.

[20] P.E. Moraal and J.W. Grizzle. Observer design for nonlinear systems with discret-time measurements. *IEEE Trans. on Aut. Control*, 40(3):395–404, 1995.
[21] M. Nadri and H. Hammouri. Design of a continuous-discrete observer for state affine systems. *Appl. Math. Lett.*, 16:967–974, 2003.

[22] M. Nadri, H. Hammouri, and C.-M. Astorga. Observer design for continuous-discrete time state affine systems up to output injection. *Eur. J. Control*, 10:252–263, 2004.

[23] Q. Zhang. Adaptive observer for multi-input-multi-output (mimo) linear time-varying systems. *IEEE Trans. on Aut. Control*, 47(3):525–529, 2002.

[24] Q. Zhang. Revisiting different adaptive observers through a unified formulation. In *Proc. of the 45th IEEE Conf. on Dec. & Control and Eur. Conf. on Control*, pages 3067–3072, Seville, Spain, 2005.
Références

[1] C. Aboky, G. Sallet, and J.-C. Vivalda. Observers for Lispschitz non-linear systems. *International Journal of Control*, 75(3):204-212, 2002.

[2] M. Alamir. Optimization based nonlinear observers revisited. *International Journal of Control*, 72(13):1204-1217, 1999.

[3] T. Ahmed Ali, R. Postoyan, and F. Lamnabhi-Lagarrigue. Continuous-discrete adaptive observers for state affine systems. In *CDC'07 (IEEE Conference on Decision and Control) New Orleans, U.S.A.*, pages 1356-1360, 2007.

[4] B.D.O. Anderson, R.R. Bitmead, C.R. Johnson, P.V. Kokotović, R.L. Kosut, I.M.Y. Mareels, L. Praly, and B.D. Riedle. *Stability of Adaptive Systems: Passivity and Average Raging Analysis*. Signal Processing, Optimization and Control, MIT Press, Cambridge, U.S.A., 1986.

[5] D. Angeli and E. Mosca. Command governors for constrained nonlinear systems. *IEEE Transactions on Automatic Control*, 44:816-818, 1999.

[6] D. Angeli and E.D. Sontag. Forward completeness, unboundedness observability, and their Lyapunov characterizations. *Systems & Control Letters*, 38:209-217, 1999.

[7] A.M. Annaswamy, F. Skantze, and A.-P. Loh. Adaptive control of continuous-time systems with convex/concave parametrization. *Automatica*, 34(1):33-49, 1998.

[8] M. Arcak and P. Kokotović. Nonlinear observers: a circle criterion design and robustness analysis. *Automatica*, 37:1923-1930, 2001.

[9] M. Arcak and D. Nešić. A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation. *Automatica*, 40:1931-1938, 2004.

[10] A. Aropostathis, A. Jakubczyk, B. Lee, H.G. Marcus, and E.D. Sontag. The effect of sampling on linear equivalence and feedback linearization. *Systems & Control Letters*, 13:373-381, 1989.

[11] K. Årzén, A. Cervin, and J. Eker. An introduction to control and scheduling co-design. In *CDC'00 (IEEE Conference on Decision and Control) Sidney, Australia*, pages 4865-4870, 2000.

[12] A. Astolfi, D. Karagiannis, and R. Ortega. *Nonlinear and Adaptive Control with Applications*. Springer, New York, U.S.A., 2008.