## 3 Main Result
### Section 1  
We consider Equation (12) with \(\tau=0\); then, by considering feedback controller \(u_{i}=k_{i}x_{i}\), the closed loop system can be rewritten as  
\[{}^{C}D_{t}^{\mathrm{p}}x_{i}=a_{i^{\mathrm{cl}}}x_{i}+f_{i}(x)+g_{i}(x_{i})+b _{i}\bar{\bar{\psi}}_{i}(k_{i}x_{i}),\mathrm{i}\in\{1,2,\ldots,N\}, \tag{13}\]  
where \(\bar{\bar{\psi}}_{i}=sat(k_{i}x_{i})-k_{i}x_{i}\) and \(a_{i^{\mathrm{cl}}}=a_{i}+b_{i}k_{i}\). We introduce some assumptions for the present analysis. Some of these relate to the functions \(f_{i}(x)\) and \(g(x_{i})\), as introduced in [28; 29]. Additionally, the following components are defined and utilized in the subsequent discussions:  
Figure 1: Block diagram of the FO delayed system subject to saturation control. 1. Function \(\Phi_{3i}(\|x_{i}\|)\): Let \(\Phi_{3i}:[0,\infty)\rightarrow[0,\infty)\) be a non-negative, continuous, and radially unbounded function that depends on the norm \(\|x_{i}\|\). This function quantifies bounds on the growth of specific terms in the analysis of subsystem \(i\). 2. Lyapunov Function Gradient \(\
abla_{x_{i}}V_{i}(x_{i})\): The term \(\
abla_{x_{i}}V_{i}(x_{i})\) represents the gradient of the Lyapunov function \(V_{i}(x_{i})\) with respect to the state \(x_{i}\). The Lyapunov function \(V_{i}(x_{i}):\mathbb{R}^{n}\rightarrow\mathbb{R}\) is a continuously differentiable, positive-definite function used to establish subsystem stability. 3. Upper Bound \(\Xi_{i}^{{}^{\prime\prime}},\Xi_{i}^{{}^{\prime}}\) and \(\Xi_{i}\): The scalar \(\Xi_{i}^{{}^{\prime\prime}},\Xi_{i}^{{}^{\prime}}\) and \(\Xi_{i}>0\) serves as an upper bound in inequalities involving \(\
abla_{x_{i}}V_{i}(x_{i})\) and related terms. This constant plays a crucial role in bounding subsystem dynamics and ensuring analytical rigor. **Assumption 2**. : _Consider a real number \(\Xi_{i}^{{}^{\prime\prime}}>0(i=1,\ldots,N)\) satisfying the following inequality:_  
\[\
abla_{x_{i}}V_{i}(x_{i})a_{cli}x_{i}\leq\sqrt{\Phi_{3i}(\|x_{i}\|)}\Xi_{i}^ {{}^{\prime\prime}}\sqrt{\Phi_{3i}(\|x_{i}\|)} \tag{14}\]  
**Assumption 3** ([28; 29]). : _Consider a real number \(\Xi_{ij}\geq 0(i,j=1,\ldots,N)\) satisfying the following inequality:_  
\[\
abla_{x_{i}}V_{i}(x_{i})f_{i}(x(t))\leq\sqrt{\Phi_{3i}(\|x_{i}\|)}\Xi_{ij} \sqrt{\Phi_{3i}(\|x_{j}\|)} \tag{15}\]  
**Assumption 4**. : _Consider a real number \(\Xi_{i}^{{}^{\prime}}>0(i=1,\ldots,N)\) satisfying the following inequality:_  
\[\
abla_{x_{i}}V_{i}(x_{i})\tilde{\psi}(x_{i}(t)\leq\sqrt{\Phi_{3i}(\|x_{i}\|)} \Xi_{i}^{{}^{\prime\prime}}\sqrt{\Phi_{3i}(\|x_{i}\|)} \tag{16}\]  
**Assumption 5** ([28; 29]). : _There is some constant \(v_{i}>0\) for \(\tau\geq 0\) such as_  
\[\
abla_{x_{i}}V_{i}(x)g(x_{i}(t-\tau))\leq v_{i}\Phi_{3i}^{1/2}(\|x_{i}\|)\Phi _{3i}^{1/2}(\|x_{i}(t-\tau)\|),i=\{1,2,3,\ldots\} \tag{17}\]  
**Assumption 6** ([28; 29]). : _There is a continuous non-decreasing function \(\zeta_{i}(\top)>\top>0\) for \(\tau\geq 0\) such as_  
\[\|g_{i}(x(t-\tau)\|\leq\zeta_{i}(\|g_{i}(x(t))\|),i=\{1,2,3,\ldots,N\} \tag{18}\]  
_According to Theorem 4 in [42] and Remark 2.10 in [43], if we consider a continuous nondecreasing function \(\zeta_{i}(\top)>\top>0\) for \(\top>0\), then it is clear that_  
\[\Phi_{3i}^{1/2}(\|x_{i}(t-\tau)\|)<\zeta_{i}\Big{(}\Phi_{3i}^{1/2}(\|x(t)\|) \Big{)},i=\{1,2,3,\ldots,N\} \tag{19}\]  
**Lemma 6**. : _if each \(V_{i},i\in\{1,2,\ldots,N\}\) is continuously differentiable and convex, then their sum will also be continuously differentiable and convex._  
Proof. : This pertains to properties of convex and continuously differentiable functions. **Theorem 2**. : _Considering the closed-loop system described by Equation (13) and assuming that Assumptions 1-5 are satisfied, the asymptotic stabilization of the closed-loop system can be achieved if the following judgement matrix \(W\) is an M-matrix:_  
\[\text{where }W=\left[ \begin{array}{cccc}w_{11}&w_{12}&\ldots&w_{1j}\\ w_{21}&w_{21}&\ldots&w_{2j}\\ \vdots&\vdots&\ldots&\vdots\\ w_{l1}&w_{i1}&\ldots&w_{lj}\end{array} \right]\]  
\[w_{ij}=\left\{