## Appendix A
### Stability Condition Derivation  
Derive the stability condition from Theorem 1. The derivation of Equation (13) proceeds as follows:  
To show that the expected moments of the zero solution are exponentially stable, it is sufficient to show that \(LV(t,x)\leq-\mu V(t,x)\), \(t\geq 0\).  
First, since \(x\in[0,1]\) and Equation (8) is a simple linear Equation.  
Thus, making \(V_{t}(t,x)=x,x\in[0,1]\), gives \(\frac{\partial V}{\partial t}=0\), \(\frac{\partial V}{\partial x}=1\), \(\frac{\partial^{2}V}{\partial x^{2}}=0\).  
When \(x\
eq 0\), \(V_{t}(t,x)\geq 0\); when \(x=0\), \(V_{t}(t,0)=0\).  
Therefore \(V_{t}(t,x)=x,x\in[0,1]\) is positive definite. This is the basis for stability analysis.  
It is only necessary to verify \(LV(t,x)\). If \(LV(t,x)\) is negative definite or semi-negative definite, then the system is stable at the equilibrium point.  
\[\begin{array}{l}LV(t,x)=\frac{\partial V}{\partial t}+\frac{\partial V}{ \partial x}f_{1}(x,t)+\frac{1}{2}\frac{\partial^{2}V}{\partial x^{2}}g_{1}(x,t) ^{2}\\ =[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y}+(1-z)N_{ Z}+(1-u)N_{U}+G_{X}]x\end{array} \tag{14}\]  
According to the zero-solution P-order moment-index stabilization condition in Theorem 1, if the lower is stabilized, \(LV(t,x)\leq\mu V(t,x)\) must also be satisfied.  
Thus, to satisfy Equation (15)  
\[\begin{cases}LV(t,x)\leq 0\\ LV(t,x)\leq-\mu V(t,x)=-\mu x\end{cases} \tag{15}\]  
Since \(x\in[0,1]\), it is only necessary to satisfy the Equation (16)  
\[LV(t,x)=[(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y} +(1-z)N_{Z}+(1-u)N_{U}+G_{X}]x\leq-\mu x \tag{16}\]  
That is \((1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1-y)N_{Y}+(1-z)N_{Z}+( 1-u)N_{U}+G_{X}\leq-1\), which is Equation (17).  
Similarly, Equations (18)-(20) can be derived.  
Then, derive the stabilization conditions, using X1 and X2 as examples.  
The approximate reduction process of Equation (17) is shown in Equation (A17)  
\[\begin{array}{l}(1+y\beta_{XY}+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+(1- y)N_{Y}+(1-z)N_{Z}+(1-u)N_{U}+G_{X}\\ =(1+z\beta_{XZ}+u\beta_{XU})E_{X}-C_{X}+\Pi_{X}+N_{Y}+(1-z)N_{Z}+(1-u)N_{U}+G_{ X}+y(\beta_{XY}E_{X}-N_{Y})\leq-1\\ y(\beta_{XY}E_{X}-N_{Y})\leq-1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{ Y}-(1-z)N_{Z}-(1-u)N_{U}-G_{X}\end{array}\] (A17)  
A categorical discussion of \((\beta_{XY}E_{X}-N_{Y})\) yields the conditions for satisfying Equation (A17) as  
**X1.**: _When_ \(x\in(0,1]\) _and_ \(\beta_{XY}E_{X}-N_{Y}>0\) _,_ \(y\leq[-1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{Y}-(1-z)N_{U}-G_{X} ]/(\beta_{XY}E_{X}-N_{Y})\) _and_ \(0\leq-1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{Y}-(1-z)N_{U}-G_{X} ]/(\beta_{XY}E_{X}-N_{Y})\) _and_ \(0\leq-1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{Y}-(1-z)N_{Z}-(1-u)N _{U}-G_{X}\) _should be satisfied._
**X2.**: _When_ \(x\in(0,1]\) and \(\beta_{XY}E_{X}-N_{Y}<0\)_,_ \(y\geq[-1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}\) _and_ \(-N_{Y}-(1-z)N_{Z}-(1-u)N_{U}-G_{X}]/(\beta_{XY}E_{X}-N_{Y})\)_and_ \(-1-(1+z\beta_{XZ}+u\beta_{XU})E_{X}+C_{X}-\Pi_{X}-N_{Y}-(1-z)N_{Z}-(1-u)N_{U}- G_{X}\leq 0\) _should be satisfied._  
Similarly, it can be deduced that Y1, Z1, and U1 hold.  
Similarly, it can be deduced that Equations (21)-(24) hold and it can be shown that the conditions X3, X4, Y2, Z2, and U2 hold.