## 2 Fundamental Concepts  
FC is an extension of classical calculus that allows the differentiation and integration of arbitrary (non-integer) order. The two most commonly used definitions are the Riemann-Liouville and Caputo fractional derivatives. The Riemann-Liouville derivative is defined as [1,3]  
\[D_{a}^{v}H(t)=\frac{1}{\Gamma(n-v)}\frac{d^{n}}{dt^{n}}\int_{a}^{t}(t-\tau)^{n -v-1}H(\tau)d\tau, \tag{1}\]  
where \(n=\lceil v\rceil\), \(0<v<n\), and \(\Gamma(. )\) is the Gamma function. This definition is suitable for theoretical analysis but has initial value problems with physical interpretations. The Caputo fractional derivative, often preferred in practical applications, is defined as [3]  
\[{}^{C}D_{a}^{v}H(t)=\frac{1}{\Gamma(n-v)}\int_{a}^{t}(t-\tau)^{n-v-1}\frac{d^ {n}H(\tau)}{d\tau^{n}}d\tau. \tag{2}\]  
* \(v\) is the order of the derivative, which can be fractional (non-integer). It characterizes the \"memory\" or \"hereditary\" properties of the system. The smaller the value of \(v\), the stronger the memory effects. * \(H(t)\) is the function to be differentiated. This could represent a physical quantity such as position, temperature, or voltage, depending on the application. * \(a\) is the lower bound of the fractional derivative and represents the starting point of the integral. Physically, this is often the initial time \(t=a\). * \(\Gamma(\cdot)\) is the Gamma function, a generalization of the factorial for non-integer values, ensuring the proper scaling of fractional operations. It is advantageous because it allows for standard initial conditions, such as \(H(0)\) and \(H^{\prime}(0)\), which are consistent with classical systems. In the entirety of this paper, we adopt the Caputo definition of the fractional derivative, as it is particularly suited for practical applications and allows for the use of standard initial conditions. If we consider a general non-linear FOS given by  
\[{}^{C}D_{t}^{v}x=H(x,t) \tag{3}\]  
where \(x\in\mathbb{R}^{n}\) represents the state of Equation (3), \(v\) is an FO derivative that includes \(v\in(0,1)\) and the function \(H:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}^{n}\) holds the condition \(H(0)=0\) and is locally Lipchitz with respect to the states, we can establish the following theorem. **Theorem 1** ([29; 34]). : _If the function \(H(x,t)\) is continuous over time, then a unique solution exists for the FOS given by (3). The solution is expressed as_  
\[x(t)=x_{0}+\int\limits_{0}^{\infty}\xi_{v}(w)\psi(w,t)dw, \tag{4}\]  
_where \(\psi(w,t)\) is the solution of the initial value problem associated with the FOS (3), defined as_  
\[\frac{\partial\psi(w,t)}{\partial t}=-w\psi(w,t)+H(x(t),t),\quad\psi(0,t)=0. \tag{5}\]  
_Here, \(\xi_{v}(w)=w^{-v}\frac{\sin(\pi v)}{\pi}\), where \(w\in(0,\infty)\)._  
Equation (4) represents the solution of the FOS (3) as an integral involving the vector-valued function \(\psi(w,t)\). To ensure a precise interpretation, the integral of \(\psi(w,t)\) is evaluated component-wise. That is, for \(\psi(w,t)=[\psi_{1}(w,t),\psi_{2}(w,t),\ldots,\psi_{n}(w,t)]^{\top}\), the integral is understood as  
\[x(t)=x_{0}+ \begin{bmatrix}\int\limits_{0}^{\infty}\xi_{v}(w)\psi_{1}(w,t)\,dw \\ \int\limits_{0}^{\infty}\xi_{v}(w)\psi_{2}(w,t)\,dw\\ \vdots\\ \int\limits_{0}^{\infty}\xi_{v}(w)\psi_{n}(w,t)\,dw\end{bmatrix} .\]  
This component-wise approach is consistent with the established methods in FC and ensures the solution is well defined in the context of vector-valued systems. The Lyapunov function is a critical tool for analyzing the stability of FOSs. Consider the system described in Equation (3). A Lyapunov function \(V(x):\mathbb{R}^{n}\to\mathbb{R}\) for this system must satisfy  
1. Positive Definiteness: \[V(x)>0\quad\text{for }x\
eq 0,\quad V(0)=0.\] (6)
2. Radial Unboundedness: \[V(x)\to\infty\quad\text{as }\|x\|\to\infty.\] (7) \[\|.\|\] denotes an arbitrary norm.