## 4 Asymptotic Solution to the Boundary-Value Problem (17)
### First-Order Formal Asymptotic Solution to Problems (26) and (27)
#### 4.2.4 Control-Theoretic Interpretation of the Boundary-Value Problem (39) and (37)
Solving the first equation in (82) subject to the first condition in (83), and using the positive definiteness of \(D_{u2}^{1/2}(0)\) and the estimate for \(f_{x}(\xi)\) in (86), we obtain, after a routine algebra,  
\[x_{1}^{0}(\xi)=-\int_{\xi}^{+\infty}\exp\big{(}-D_{u,2}^{1/2}(0)(\tau-\xi) \big{)}f_{x}(\tau)d\tau,\quad\xi\geq 0. \tag{87}\]  
This equation, along with Equation (84), yields  
\[y_{1}^{0}(0)=2D_{u,2}^{1/2}(0)\bigg{(}-z_{2,1}(0)+\int_{0}^{+\infty}\exp\big{(} -D_{u,2}^{1/2}(0)\tau\big{)}f_{x}(\tau)d\tau\bigg{)}. \tag{88}\]  
Thus, the second equation in (82) subject to the initial condition (88) yields the solution  
\[\frac{dy_{1}^{0}(\xi)}{d\xi}=-D_{u,2}^{1/2}(0)y_{1}^{0}(\xi)+f_{y}(\xi),\quad \xi\geq 0, \tag{82}\]0  
Due to the positive definiteness of the matrix \(D_{u,2}^{1/2}(0)\) and the estimate for \(f_{y}(\xi)\) (see Equation (86)), the vector-valued function \(y_{1}^{0}(\xi)\), given by (89), satisfies the limit condition in (83). Now, using Equations (81) and (87) and (89), we obtain the solution to System (75) subject to Conditions (77) and (79)  
\[\frac{dy_{1}^{0}(\xi)}{d\xi}=-D_{u,2}^{1/2}(0)y_{1}^{0}(\xi)+f_{y}(\xi),\quad \xi\geq 0, \tag{82}\]1 \[\frac{dy_{1}^{0}(\xi)}{d\xi}=-D_{u,2}^{1/2}(0)y_{1}^{0}(\xi)+f_{y}(\xi),\quad \xi\geq 0, \tag{82}\]2  
Due to the positive definiteness of the matrix \(D_{u,2}^{1/2}(0)\), the estimates in (86) and Equations (87) and (89), this solution satisfies the inequalities  
\[\frac{dy_{1}^{0}(\xi)}{d\xi}=-D_{u,2}^{1/2}(0)y_{1}^{0}(\xi)+f_{y}(\xi),\quad \xi\geq 0, \tag{82}\]3  
where \(\gamma_{2}^{0}>0\) and \(\gamma_{\lambda}^{0}>0\) are some constants. Proceed to the solution to System (76) subject to Conditions (78) and (80). Similarly to (81), we make the following transformation of the state variables in Problems (76), (78), and (80):  
\[\left( \begin{array}{c}z_{2,1}^{f}(\eta)\\ \lambda_{u2,1}^{f}(\eta)\end{array} \right)=\Theta(t_{f})\left( \begin{array}{c }x_{1}^{f}(\eta)\\ y_{1}^{f}(\eta)\end{array} \right), \tag{92}\]  
where \(x_{1}^{f}(\eta)\in E^{r}\) and \(y_{1}^{f}(\eta)\in E^{r}\) are new state variables. Transformation (92) converts Problems (76), (78), and (80) to the problem  
\[\frac{dx_{1}^{f}(\eta)}{d\eta}=D_{\mathrm{u},2}^{1/2}(t_{f})x_{1}^{f} (\eta),\quad\eta\leq 0,\] \[\frac{dy_{1}^{f}(\eta)}{d\eta}=-D_{\mathrm{u},2}^{1/2}(t_{f})y_{1} ^{f}(\eta),\quad\eta\leq 0, \tag{93}\] \[\lim_{\eta\to-\infty}x_{1}^{f}(\eta)=0,\quad\lim_{\eta\to-\infty} y_{1}^{f}(\eta)=0,\] (94) \[x_{1}^{f}(0)=0.5z_{2,1}^{f}(0)+0.5D_{\mathrm{u},2}^{-1/2}(t_{f} )\bar{\lambda}_{\mathrm{u},2,1}(t_{f}),\] \[y_{1}^{f}(0)=D_{\mathrm{u},2}^{1/2}(t_{f})z_{2,1}^{f}(0)-\bar{ \lambda}_{\mathrm{u},2,1}(t_{f}). \tag{95}\]  
The second equation in (93) yields (subject to the second condition in (95))  
\[y_{1}^{f}(\eta)=\exp\big{(}-D_{\mathrm{u},2}^{1/2}(t_{f})\eta\big{)}y_{1}^{f} (0),\quad\eta\leq 0. \tag{96}\]  
Since the matrix \(D_{\mathrm{u},2}^{1/2}(t_{f})\) is positive definite, then the vector-valued function \(y_{1}^{f}(\eta)\), obtained in (96), satisfies the second limit equality in (94) if and only if \(y_{1}^{f}(0)=0\). The latter, along with Equation (95), yields  
\[x_{1}^{f}(0)=D_{\mathrm{u},2}^{-1/2}(t_{f})\bar{\lambda}_{\mathrm{u},2,1}(t_{ f}). \tag{97}\]  
Hence, the solution to the second equation in (93) subject to the aforementioned condition \(y_{1}^{f}(0)=0\), is identically zero, i.e.,  
\[y_{1}^{f}(\eta)\equiv 0,\quad\eta\leq 0. \tag{98}\]  
Solving the first equation in (93) subject to the condition obtained in (97), we have  
\[x_{1}^{f}(\eta)=\exp\big{(}D_{\mathrm{u},2}^{1/2}(t_{f})\eta\big{)}D_{ \mathrm{u},2}^{-1/2}(t_{f})\bar{\lambda}_{\mathrm{u},2,1}(t_{f}),\quad\eta \leq 0. \tag{99}\]  
Since the matrix \(D_{\mathrm{u},2}^{1/2}(t_{f})\) is positive definite, then the vector-valued function \(x_{1}^{f}(\eta)\), obtained in (99), satisfies the first limit equality in (94). Thus, the vector-valued functions \(x_{1}^{f}(\eta)\) and \(y_{1}^{f}(\eta)\), obtained in (99) and (98), respectively, constitute the solution to Problems (93)-(95).