## 3 Detection of GFPS Condition Based on the Laplace Transform
\begin{bmatrix}kI_{n}&0&0\\ \alpha_{1}&\alpha_{2}&0\\ \alpha_{3}&0&\alpha_{4}\end{bmatrix} , \tag{13}\]  
\(I_{n}\), \(I_{m}\), \(I_{l}\) represent the \(n\times n\), \(m\times m\) and \(l\times l\) real identity matrices, respectively. \(T\) denotes the transpose of vectors. \(e_{j0}\), \(j=1,2,3\), are given initial values of system (10) and \(\hat{F}_{1,2,3}\) are the Laplace transforms of \(F_{1,2,3}\), respectively, i.e.,  
\[\hat{F}_{j}=L[F_{j}]=\int_{0}^{+\infty}F_{j}(t)e^{-st}dt,\quad j=1,2,3.\]  
The solution to Equation (12) can be derived by using Cramer's rule [48]. Take, for example, the expression of \(E_{1}\), which is an \(n\)-dimensional vector and always can be given by the form below:  
\[E_{1}= \begin{bmatrix}E_{11}\\ E_{12}\\ \vdots\\ E_{1n}\end{bmatrix} =\frac{1}{D(s)} \begin{bmatrix}w_{11}(s)\\ w_{12}(s)\\ \vdots\\ w_{1n}(s)\end{bmatrix} +\frac{1}{D(s)} \begin{bmatrix}w_{21}(s)\hat{F}_{11}(s)\\ w_{22}(s)\hat{F}_{12}(s)\\ \vdots\\ w_{2n}(s)\hat{F}_{1n}(s)\end{bmatrix} , \tag{14}\]  
where \((E_{11},E_{12},\cdots,E_{1n})\) and \((\hat{F}_{11},\hat{F}_{12},\cdots,\hat{F}_{1n})\) are the \(n\) components of the vectors \(E_{1}\) and \(\hat{F}_{1}\), respectively. \(D(s)\) is the determinant of matrix \((S-M)\). \(w_{1j},w_{2j}\), \(j=1,\cdots,n\), are polynomials of \(s\), and their highest powers are less than that of \(D(s)\). Taking the inverse Laplace transform on Equation (14) and applying the convolution theorem, one has  
\[e_{1j}=L^{-1}[E_{1j}]=L^{-1}[\frac{w_{1j}(s)}{D(s)}]+L^{-1}[\frac{w_{2j}(s)}{D (s)}]*F_{1j}(t),\quad j=1,\cdots,n. \tag{15}\]  
where \((e_{11},e_{12},\cdots,e_{1n})\) and \((F_{11},F_{12},\cdots,F_{1n})\) are the \(n\) components of the vectors \(e_{1}\) and \(F_{1}\), respectively. \(*\) denotes the convolution operation. First, we introduce the following theorem. Theorem 3.1: _The necessary condition for \(e_{1j}\to 0\) with \(t\to\infty\), \(j=1,2,\cdots,n\), in Equation (15) is that all eigenvalues of matrix \(M\) defined in Equation (13) have negative real parts._  
Proof: Without loss of generality, for any \(i=1,2\) and \(j=1,2,\cdots,n\), \(w_{ij}(s)/D(s)\) have the following form  
\[\frac{w_{ij}(s)}{D(s)}=\frac{\sum\limits_{p=0}^{v}a_{p}s^{p}}{\sum\limits_{p= 0}^{n+m+l}b_{p}s^{p}}, \tag{16}\]  
where \(v<n+m+l\), \(a_{p}\) and \(b_{p}\) are constants. Consider the following three cases of the inverse Laplace transforms for Equation (16):\(D(s)\) has \(n+m+l\) single roots \(s_{1},s_{2},\cdots,s_{(n+m+l)}\) \(\frac{w_{ij}(s)}{D(s)}=\frac{q_{1}}{s-s_{1}}+\frac{q_{2}}{s-s_{2}}+\cdots+\frac {q_{(n+m+l)}}{s-s_{(n+m+l)}}\), where \(q_{p}=\frac{w_{ij}(s)}{D(s)}(s-s_{p})\Big{|}_{s=s_{p}}\), \(p=1,2,\cdots,n+m+l\). \(L^{-1}[\frac{w_{ij}(s)}{D(s)}]=q_{1}e^{s_{1}t}+q_{2}e^{s_{2}t}+\cdots+q_{(n+m +l)}e^{s_{(n+m+l)}t}\) \(D(s)\) has a pair of conjugate complex roots \(s_{1,2}=\beta\pm j\omega\) \(\frac{w_{ij}(s)}{D(s)}=\frac{q_{ij}(s)}{(s-\beta-\omega)(s-\beta+j\omega)Q(s)} =\frac{q_{1}}{s-\beta-\omega}+\frac{q_{2}}{s+j\omega}+\frac{P(s)}{Q(s)}\), where \(q_{1}=\frac{w_{ij}(s)}{D^{(s)}}\Big{|}_{s=\beta+j\omega}\), \(q_{2}=\frac{w_{ij}(s)}{D^{(s)}}\Big{|}_{s=\beta-j\omega}\), \(L^{-1}[\frac{w_{ij}(s)}{D(s)}]=q_{1}e^{(\beta+j\omega)t}+q_{2}e^{(\beta-j\omega )t}+L^{-1}[\frac{P(s)}{Q(s)}]\) \(D(s)\) has \((n+m+l)\)-repeated real roots \(s=s_{0}\) \(\frac{w_{ij}(s)}{D(s)}=\frac{q_{1}}{s-s_{0}}+\frac{q_{2}}{(s-s_{0})^{2}}+\cdots +\frac{q_{(n+m+l)}}{(s-s_{0})^{n+m+l)}}\), where \(q_{(n+m+l-p)}=\frac{1}{p!}\frac{d!}{d! }\frac{w_{ij}(s  
Proof. : In fact, Equations (17) and (18) are a series of Volterra integral equations, which can be solved by using the successive approximation method [47] in the theory of integral equations. Consider the Volterra integral equation as below:  
\[A(t)=B(t)+\int_{0}^{t}T(t-\tau)H(\tau,A(\tau))d\tau, \tag{19}\]  
where \(A(t)\in R^{n}\) and \(|A(t)|<\infty\), \(T(t)\) is a \(n\times n\) matrix, and \(|T(t)|\in L[0,t_{0}]\) for any \(0<t_{0}<\infty\). Both \(B(t)\) and \(H(t,A(t))\) are vectors with \(n\) continuous components.