## 3 Main Result
### Section 1
The system is guaranteed to be asymptotically stable if there exists_  
* _A symmetric positive-definite matrix_ \(P\) _with all elements satisfying_ \(p_{ij}\geq 0\)_,_
* _Positive constants_ \(k_{f}\)_,_ \(k_{g}\)_, and_ \(k_{s}\)_,_  
_such that the following matrix inequality holds:_  
\[\left[ \begin{array}{cccc}\Theta&P&P&P\\ P&-k_{f}I&0&0\\ P&0&-k_{g}I&0\\ P&0&0&-k_{s}I\end{array} \right]<0, \tag{29}\]  
_where_  
\[\Theta=PA_{cl}+A_{cl}^{T}P+2k_{f}l_{1}^{2}+2k_{g}l_{2}^{2}+2k_{s}l_{s}^{2}.\]  
Proof. : In this proof, we use Lyapunov-like function method (often used to establish stability in control theory for FOS), and it contains a dependence on both the state \(x(t)\) and the time \(t\) because the solution \(\psi(w,t)\) depends on \(t\). By introducing the following Lyapunov function based on Theorem 1, we have  
\[V(x,t)=\int\limits_{0}^{\infty}\xi_{v}(w)\psi(w,t)^{T}P\psi(w,t)dw \tag{30}\]  
if we obtain the time derivative from \(V(x)\), it is clear that  
\[\dot{V}(x,t)=\int\limits_{0}^{\infty}\xi_{v}(w)\Bigg{[}\frac{\partial\psi(w,t )}{\partial t}^{T}P\psi(w,t)+\psi(w,t)^{T}P\frac{\partial\psi}{\partial t} \Bigg{]}dw \tag{31}\]  
Using Theorem 1, the above equation can be rewritten  
\[ \begin{array}{l}\dot{V}(x,t)=\int\limits_{0}^{\infty}\xi_{v}(w)[-w\psi(w,t) +A_{cl}x+F(x)+G(x)+B\Psi(x)]^{T}P\psi(w,t)dw\\ +\int\limits_{0}^{\infty}\xi_{v}(w)\psi^{T}(w,t)P[-w\psi(w,t)+A_{cl}x+F(x)+G( x)+B\Psi(x)]dw\end{array}  \tag{32}\]  
By simplifying, we have  
\[ \begin{array}{l}\dot{V}(x,t)=-\int\limits_{0}^{\infty}\xi_{v}(w)\psi^{T}(w,t )P\psi(w,t)dw+x^{T}A_{cl}^{T}Px+F^{T}Px+G^{T}Px+\Psi^{T}(x)B^{T}Px\\ -\int\limits_{0}^{\infty}\xi_{v}(w)\psi^{T}(w,t)P\psi(w,t)dw+x^{T}PA_{cl}x+x^{ T}PF+x^{T}PG(x)+x^{T}PB\Psi(x)\end{array}  \tag{33}\]Then, based on the Lipschitz condition, for any positive constant \(k_{f},k_{g}>0\) and \(k_{s}\),  
\[ \begin{array}{l}k_{f}(F^{T}(x(t))F(x(t))-F^{T}(x(t))F(x(0))-F(x(t)F^{T}(x(0))+F^ {T}(x(0))F(x(0))\\ \leq 2k_{f}l_{1}^{2}(x^{T}(t)x(t)-x^{T}(t)x(0)-x(t)x^{T}(0)\\ +x^{T}(0)x(0)2k_{f}l_{2}^{2}x^{T}(t)x(t)-k_{f}F^{T}(x(t))F(x(t))>0\end{array}  \tag{34}\]  
where \(l_{1}\) is the Lipschitz parameter of \(F(.)\)[29]. \[ \begin{array}{l}k_{g}(G^{T}(x(t))G(x(t))-G^{T}(x(t))G(x(0))-G(x(t)G^{T}(x(0))+ G^{T}(x(0))G(x(0))\\ \leq 2k_{g}l_{2}^{2}(x^{T}(t)x(t)-x^{T}(t)x(0)-x(t)x^{T}(0)\\ +x^{T}(0)x(0)2k_{g}l_{2}^{2}x^{T}(t)x(t)-k_{g}G^{T}(x(t))G(x(t))>0\end{array}  \tag{35}\]  
where \(l_{2}\) is the Lipchitz parameter of \(G(.)\). \[ \begin{array}{l}k_{s}(\Psi^{T}(x(t))\[PERSON](x(t))-\Psi^{T}(x(t))\Psi(x(0))-\Psi (x(t)\Psi^{T}(x(0))+\Psi^{T}(x(0))\Psi(x(0))\\ \leq 2k_{g}l_{s}^{2}(x^{T}(t)x(t)-x^{T}(t)x(0)-x(t)x^{T}(0)\\ +x^{T}(0)x(0)2k_{g}l_{s}^{2}x^{T}(t)x(t)-k_{s}\Psi^{T}(x(t))\Psi(x(t))>0\end{array}  \tag{36}\]  
where \(l_{s}\) is the Lipchitz parameter of \(\Psi(.)\). Then, we can continue as follows  
\[ \begin{array}{l}\tilde{V}(x,t)=-2\int\limits_{0}^{\infty}\xi_{v}(w)\psi^{T} (w,t)P\psi(w,t)dw+x^{T}(PA_{cl}+A_{cl}^{T}P)x+x^{T}Pf(x)\\ +x^{T}Pg(x)+x^{T}P\Psi(x)+f^{T}(x)Px+g^{T}Px+\Psi^{T}(x)B^{T}Px+2k_{f}l_{1}^ {2}x^{T}x\\ -k_{f}f^{T}(x)f(x)+2k_{g}l_{2}^{2}x^{T}x-k_{g}g^{T}(x)g(x)+2k_{g}l_{s}^{2}x^{ T}x-k_{s}\Psi^{T}(x)\Psi(x)\end{array}  \tag{37}\]  
By definition of a new vector as \(\Xi=\left[ \begin{array}{cccc}x^{T}&f^{T}&g^{T}&\psi^{T}\end{array} \right]^{T}\), we have  
\[\Xi^{T}\left[ \begin{array}{cccc}\Theta&P&P&P\\ P&-k_{f}I&0&0\\ P&0&-k_{g}I&0\\ P&0&0&-k_{g}I\end{array} \right]\Xi<0 \tag{38}\]  
Then, the FO non-linear system subject to saturation is asymptotically stable, which completes the proof. The stability conditions provided by Theorem 3 offer valuable insights into the design of controllers and stability analysis for the considered system. By ensuring that the matrix inequality (29) is satisfied, we can guarantee the stability of the system, enabling effective control strategies.