## 4 Asymptotic Solution to the Boundary-Value Problem (17)
### First-Order Formal Asymptotic Solution to Problems (26) and (27)
By virtue of the Boundary Functions Method, these conditions are\[\lim_{\xi\rightarrow+\infty}z_{1,0}^{0}(\xi)=0,\ \ \lim_{\xi\rightarrow+\infty} \lambda_{\text{u1,0}}^{0}(\xi)=0,\ \ \lim_{\xi\rightarrow+\infty}\lambda_{\text{v,0}}^{0}(\xi)=0,\ \ \lim_{\xi\rightarrow+\infty}\lambda_{\text{u,v,0}}^{0}(\xi)=0. \tag{31}\]  
The set of Equations (30) subject to Conditions (31) has the unique solution  
\[z_{1,0}^{0}(\xi)\equiv 0,\ \ \lambda_{\text{u1,0}}^{0}(\xi)\equiv 0,\ \ \lambda_{\text{v,0}}^{0}(\xi)\equiv 0,\ \ \lambda_{\text{u,v,0}}^{0}(\xi)\equiv 0,\ \ \xi\geq 0. \tag{32}\]  
2.2 Obtaining the Boundary Corrections \(z_{1,0}^{f}(\eta)\), \(\lambda_{\text{u1,0}}^{f}(\eta)\), \(\lambda_{\text{v,0}}^{f}(\eta)\) and \(\lambda_{\text{u,v,0}}^{f}(\eta)\)  
For these boundary corrections, we obtain the following equations:  
\[\frac{dz_{1,0}^{f}(\eta)}{d\eta}=0,\ \ \ \eta\leq 0,\] \[\frac{d\lambda_{\text{u1,0}}^{f}(\eta)}{d\eta}=0,\ \ \ \eta\leq 0,\] \[\frac{d\lambda_{\text{v,0}}^{f}(\eta)}{d\eta}=0,\ \ \ \eta\leq 0, \tag{33}\] \[\frac{d\lambda_{\text{u,v0}}^{f}(\eta)}{d\eta}=0,\ \ \ \eta\leq 0.\]  
Due to the Boundary Functions Method, we require that  
\[\lim_{\eta\rightarrow-\infty}z_{1,0}^{f}(\eta)=0,\ \ \lim_{\eta\rightarrow-\infty} \lambda_{\text{u1,0}}^{f}(\eta)=0,\ \ \lim_{\eta\rightarrow-\infty}\lambda_{\text{v,0}}^{f}(\eta)=0,\ \ \lim_{\eta\rightarrow-\infty}\lambda_{\text{u,v,0}}^{f}(\eta)=0. \tag{34}\]  
Thus, the set of Equations (33) subject to Conditions (34) yields the unique solution  
\[z_{1,0}^{f}(\eta)\equiv 0,\ \ \lambda_{\text{u1,0}}^{f}(\eta)\equiv 0,\ \ \lambda_{\text{v,0}}^{f}(\eta)\equiv 0,\ \ \lambda_{\text{u,v,0}}^{f}(\eta)\equiv 0,\ \ \ \eta\leq 0. \tag{35}\]  
2.3 Obtaining the Outer Solution Terms \(\bar{z}_{1,0}(t)\), \(\bar{z}_{2,0}(t)\), \(\bar{\lambda}_{\text{u1,0}}(t)\), \(\bar{\lambda}_{\text{u2,0}}(t)\), \(\bar{\lambda}_{\text{v,0}}(t)\), \(\bar{\lambda}_{\text{u,v,0}}(t)\)  
Due to Remark 5 and Equations (32) and (35), the outer solution's terms \(\bar{z}_{1,0}(t)\), \(\bar{z}_{2,0}(t)\), \(\bar{\lambda}_{\text{u1,0}}(t)\), \(\bar{\lambda}_{\text{u2,0}}(t)\), \(\bar{\lambda}_{\text{u,v,0}}(t)\), \(\bar{\lambda}_{\text{u,v,0}}(t)\) satisfy the following system of equations in the time interval \([0,t_{f}]\):  
\[\frac{d\bar{z}_{1,0}(t)}{dt}=A_{1}(t)\bar{z}_{1,0}(t)+A_{2}(t) \bar{z}_{2,0}(t)-S_{\text{v,1}}(t)\bar{\lambda}_{\text{v,0}}(t),\] \[\bar{\lambda}_{\text{u2,0}}(t)=0,\] \[z_{1,0}^{0}(\xi)\equiv 0,\ \ \lambda_{\text{u1,0}}^{0}(\xi)\equiv 0,\ \ \lambda_{\text{v,0}}^{0}(\xi)\equiv 0,\ \ \lambda_{\text{u,v,0}}^{0}(\xi)\equiv 0,\ \ \xi\geq 0. \tag{32}\]0 \[z_{1,0}^{0}(\xi)\equiv 0,\ \ \lambda_{\text{u1,0}}^{0}(\xi)\equiv 0,\ \ \lambda_{\text{v,0}}^{0}(\xi)\equiv 0,\ \ \lambda_{\text{u,v,0}}^{0}(\xi)\equiv 0,\ \ \xi\geq 0. \tag{32}\]1 \[\frac{d\bar{\lambda}_{\text{v,0}}(t)}{dt}=-D_{\text{v}}(t)\left( \begin{array}{c}\bar{z}_{1,0}(t)\\ \bar{z}_{2,0}(t)\end{array} \right)-A^{T}(t)\bar{\lambda}_{\text{v,0}}(t),\] \[\frac{d\bar{\lambda}_{\text{u,v,0}}(t)}{dt}=S_{\text{v}}(t)\left( \begin{array}{c}\bar{\lambda}_{\text{u1,0}}(t)\\ 0\end{array} \right)-S_{\text{u,v}}(t)\bar{\lambda}_{\text{v,0}}(t)+A(t)\bar{ \lambda}_{\text{u,v,0}}(t), \tag{36}\]  
\[z_{1,0}(0)=z_{01},\ \ \ \bar{\lambda}_{\text{u1,0}}(t_{f})=0,\ \ \ \bar{\lambda}_{\text{v,0}}(t_{f})=0,\ \ \ \bar{\lambda}_{\text{u,v,0}}(0)=0. \tag{37}\]  
**Remark 6**. : _System (36) consists of four differential equations with respect to \(\bar{z}_{1,0}(t)\), \(\bar{\lambda}_{\text{u1,0}}(t)\), \(\bar{\lambda}_{\text{v,0}}(t)\), and \(\bar{\lambda}_{\text{u,v,0}}(t)\), and two algebraic equations with respect to \(\bar{\lambda}_{\text{u2,0}}(t)\) and \(\bar{z}_{2,0}(t)\).