## 3 Main Result
### Section 1
\begin{array}{ll}-\Xi_{i}^{{}^{\prime\prime}}-\zeta_{i}v_{i} -\|b_{i}\|\Xi_{i}^{{}^{\prime}}&\text{if }\ i=j\\ -\Xi_{ij}&\text{Otherwise}\end{array} \right. \tag{20}\]Proof. : Let us consider a series of positive-definite convex and continuously differentiable functions \(V_{i}(x_{i})\) that satisfy Lemma 1 for the non-linear FOS. \[ \begin{array}{l}\alpha_{1i}(\|x_{i}\|)\leq V_{i}\leq\alpha_{2i}(\|x_{i}\|)\\ {}^{C}D^{\mathrm{v}}V(x_{i})\leq-\alpha_{3i}(\|x_{i}\|)\end{array}  \tag{21}\]  
Then, we chose a vector function, the Lyapunov function of the FOS, with a set of positive-definite constants \(q_{i}>0\). \[V(x)=\sum\limits_{i=1}^{N}q_{i}V_{i}(x_{i}) \tag{22}\]  
According to Lemma 2, we have  
\[{}^{C}D^{\mathrm{v}}V(x)\leq\sum\limits_{i=1}^{N}q_{i}\
abla_{x_{i}}V_{i}(x_{i })^{C}D^{\mathrm{v}}x_{i} \tag{23}\]  
Substituting the states equation of (13) in (22), we have  
\[{}^{C}D^{\mathrm{v}}V(x)\leq\sum\limits_{i=1}^{N}q_{i}\
abla_{x_{i}}V_{i}(x_{ i})(a_{ci}x_{i}+f_{i}(x)+g_{i}(x(t))+b_{i}\bar{\psi}_{i}(k_{i}x_{i})),i\in\{1,2, \ldots,N\}, \tag{24}\]  
Simplifying Equation (24), we have  
\[ \begin{array}{l}{}^{C}D^{\mathrm{v}}V(x)\leq\sum\limits_{i=1}^{N}q_{i}( \
abla_{x_{i}}V_{i}(x_{i})a_{ci}x_{i}+\
abla_{x_{i}}V_{i}(x_{i})f_{i}(x)+\\ \
abla_{x_{i}}V_{i}(x_{i})g_{i}(x(t))+\
abla_{x_{i}}V_{i}(x_{i})b_{i}\bar{\psi }_{i}(k_{i}x_{i})),i\in\{1,2,\ldots,N\}\end{array}  \tag{25}\]  
Applying Equations (14), (15), (17) and (16),  
\[ \begin{array}{l}{}^{C}D^{\mathrm{v}}V(x)\leq\sum\limits_{i=1}^{N}q_{i}( \sqrt{\[PERSON]_{3i}(\|x_{i}\|)}\[PERSON]_{i}^{\mathrm{v}}\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_{i}(t)\|) }\Big{)}+b_{i}\sqrt{\[PERSON]_{3i}(\|x_{i}\|)}\[PERSON]_{i}^{\prime}\sqrt{\[PERSON]_{3i}(\|x_ {i}\|)}\\,i\in\{1,2,\ldots,N\}\end{array}  \tag{26}\]  
Simplifying the above equation, we have  
\[{}^{C}D^{\mathrm{v}}V(x)\leq-\frac{1}{2}\[PERSON]_{3}^{\mathrm{T}}(\|x\|)(WQ+QW) \[PERSON]_{3}(\|x\|) \tag{27}\]  
we are able to find a diagonal matrix \(Q=diag\{q_{1},q_{2},...,q_{N}\}\), such that the matrix \(WQ+WQ\) is positive-definite. Then, according to Lemma 4, \(W\) is an M-matrix. Thus, based on Lemma 3, the closed-loop system is asymptotically stable. \(\quad\) where \(\Phi_{3}(\|x\|)=\left[ \begin{array}{cccc}\Phi_{31}^{\frac{1}{2}}&\Phi_{32}^ {\frac{1}{2}}&\ldots&\Phi_{3N}^{\frac{1}{2}}\end{array} \right]\). The proof is complete. By satisfying the M-matrix properties of \(W\), we can guarantee the stability and asymptotic stabilization of the closed-loop system described by Equation (13). This provides a basis for designing effective stabilization controllers that ensure the convergence of the system's trajectories to the desired equilibrium point. We could introduce an alternative theorem in order to check a new stability condition by the Lyapunov-like function with the aid of diffusive realization and the Lipchitz condition. The Lyapunov-like function method provides a powerful tool for analyzing the stability of non-linear FOSs. Then, the described nonlinear system with Equation (13) can be rewritten as general form as  
\[{}^{C}D_{t}^{n}x=A_{cl}x+F(x)+G(x(t))+B\Psi(x), \tag{28}\]  
where \(A_{cl}=diag\{a_{cl1},a_{cl2},...,a_{clN}\}\) and \(B=diag\{b_{1},b_{2},\ldots,b_{N}\}\) are constant matrices, and \(F(x)\), \(G(x(t)\) are the matrix of the linear or non-linear part of the system with appropriate dimension. Furthermore, \(\Psi(x)=Sat(Kx)-Kx\), \(K=diag\{k_{1},k_{2},\ldots,k_{N}\}\). **Lemma 7** ([19]). : _By defining function \(\Psi(x)=Sat(u)-u\), it satisfies the Lipchitz condition with the Lipchitz parameter of \(l_{s}\)._  
**Theorem 3**. : _Consider the dynamical system described by Equation (28) under the validity of Assumption 1.