## 3 Main Result
### Section 2  
The second analysis considers the presence of a time delay, indicated by \(\tau\
eq 0\) in Equation (12). Then, Equation (39) presents the closed-loop system:  
\[{}^{C}D_{t}^{\varphi}x_{i}=a_{,i}x_{i}+f_{i}(x)+g_{i}(x(t-\tau))+b_{i}\vec{ \psi}_{i}(k_{i}x_{i}),i\in\{1,2,\ldots,N\}, \tag{39}\]  
**Theorem 4**. : _Consider the closed-loop system described by Equation (39), which satisfies Assumptions 2-6. If the judgment matrix \(W\) is an M-matrix, then the closed-loop system (39) is asymptotically stable.__The judgment matrix \(W\) is defined as_  
\[W= \begin{bmatrix}w_{11}&w_{12}&\cdots&w_{1j}\\ w_{21}&w_{22}&\cdots&w_{2j}\\ \vdots&\vdots&\ddots&\vdots\\ w_{i1}&w_{i2}&\cdots&w_{ij}\end{bmatrix} ,\]  
_where the elements \(w_{ij}\) of \(W\) are given by_  
\[w_{ij}= \begin{cases}-\Xi_{i}^{\prime\prime}-\zeta_{i}v_{i}-\|b_{i}\|\Xi_{i}^{ \prime}&\text{if }i=j,\\ -\Xi_{ij}&\text{otherwise}.\end{cases}  \tag{40}\]  
_This theorem establishes that if the matrix \(W\) satisfies the properties of an \(M\)-matrix, the stability of the closed-loop system is guaranteed. The conditions on the elements \(w_{ij}\) ensure that the system's stability criteria are met even in the presence of nonlinearities and external constraints._  
Proof. : For a closed-loop system (39), similar to the previous proof, let us consider a series of positive-definite functions \(V_{i}(x_{i})\) as the Lyapanov function that satisfies Lemma 1 such as  
\[V(x)=\sum\limits_{i=1}^{N}q_{i}V_{i}(x_{i}) \tag{41}\]  
where \(q_{i}\) is a set of positive-definite constants. Then, according to Lemma 2, the following inequality satisfies  
\[C\mathcal{D}^{v}V(x)\leq\sum\limits_{i=1}^{N}q_{i}\
abla_{x_{i}}V_{i}(x_{i})^ {C}D^{v}x_{i} \tag{42}\]  
Substituting the states equation of (39) in (42), we have  
\[C\mathcal{D}^{v}V(x)\leq\sum\limits_{i=1}^{N}q_{i}\
abla_{x_{i}}V_{i}(x_{i})( a_{{}_{d}i}x_{i}+f_{i}(x)+g_{i}(x(t-\tau))+b_{i}\bar{\psi}_{i}(k_{i}x_{i})),i \in\{1,2,\ldots,N\}, \tag{43}\]  
Simplifying Equation (43), we have  
\[ \begin{split} C\mathcal{D}^{v}V(x)\leq\sum\limits_{i=1}^{N}q_{i}( \
abla_{x_{i}}V_{i}(x_{i})a_{{}_{d}i}x_{i}+\
abla_{x_{i}}V_{i}(x_{i})f_{i}(x)+ \\ \
abla_{x_{i}}V_{i}(x_{i})g_{i}(x(t-\tau))+\
abla_{x_{i}}V_{i}(x_{i})b_{i} \bar{\psi}_{i}(k_{i}x_{i})),i\in\{1,2,\ldots,N\},\end{split}  \tag{44}\]  
Using (15) and (17)  
\[ \begin{split} C\mathcal{D}^{v}V(x)\leq\sum\limits_{i=1}^{N}q_{i}( \
abla_{x_{i}}V_{i}(x_{i})a_{{}_{d}i}x_{i}+\sqrt{\Phi_{3i}(\|x_{i}\|)}\Xi_{i} \sqrt{\Phi_{3i}(\|x_{i}\|)}+\\ v_{i}\Phi_{3i}^{1/2}(\|x_{i}\|)\Phi_{3i}^{1/2}(\|x_{i}(t-\tau)\|)+ \
abla_{x_{i}}V_{i}(x_{i})b_{i}\bar{\psi}_{i}(k_{i}x_{i})),i\in\{1,2,\ldots,N\},\end{split}  \tag{45}\]  
Applying Assumption 6 to Equation (45),  
\[ \begin{split} C\mathcal{D}^{v}V(x)\leq\sum\limits_{i=1}^{N}q_{i} \sqrt{\[PERSON]_{3i}(\|x_{i}\|)}\Xi_{i}^{\prime\prime}\sqrt{\[PERSON]_{3i}(\|x_{i}\|)} +\sqrt{\[PERSON]_{3i}(\|x_{i}\|)}\[PERSON]_{i}\sqrt{\[PERSON]_{3i}(\|x_{i}\|)}\\ +v_{i}\sqrt{\[PERSON]_{3i}(\|x_{i}\|)}\zeta_{i}\Big{(}\sqrt{\[PERSON]_{3i}(\|x(t)\|)} \Big{)}+b_{i}\sqrt{\[PERSON]_{3i}(\|x_{i}\|)}\[PERSON]_{i}^{\prime}\sqrt{\[PERSON]_{3i}(\|x_ {i}\|)}\\ {}_{i}\in\{1,2,\ldots,N\},\end{split}  \tag{46}\]Simplifying the above equation, we have  
\[{}^{C}D^{\circ}V(x)\leq-\frac{1}{2}\[PERSON]_{3}^{T}(\|x\|)(WQ+QW)\[PERSON]_{3}(\|x\|) \tag{47}\]  
we are able to always find a diagonal matrix \(Q=diag\{q_{1},q_{2},\ldots,q_{N}\}\), such that the matrix \(WQ+WQ\) is positive-definite. Then, according to Lemma 4, \(W\) is M-matrix. Thus, based on Lemma 3, the closed-loop system is asymptotically stable. \(\text{where }\Phi_{3}(\|x\|)=\left[ \begin{array}{cccc}\Phi_{31}^{\frac{1}{2}}& \Phi_{32}^{\frac{1}{2}}&\ldots&\Phi_{3N}^{\frac{1}{2}}\end{array} \right]\). Hence, the proof is complete. The selection of an appropriate vector Lyapunov function is of paramount importance in the stability analysis of the system. It serves as the foundation for ensuring that all the required assumptions are satisfied.