## 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)
From the second equation of System (70), we directly have the expression for \(\bar{\lambda}_{\mathrm{u}2,1}(t)\)  
\[\bar{\lambda}_{\mathrm{u}2,1}(t)=-\frac{d\bar{z}_{2,0}(t)}{dt}+A_{3}(t)\bar{z }_{1,0}(t)+A_{4}(t)\bar{z}_{2,0}(t)-S_{\mathrm{v}2}(t)\bar{\lambda}_{\mathrm{ v}0}(t),\quad t\in[0,t_{f}]. \tag{72}\]  
Furthermore, resolving the fourth equation in System (70) with respect to \(\bar{z}_{2,1}(t)\) and taking into account the invertibility of the matrix \(D_{\mathrm{u}2}(t)\) for all \(t\in[0,t_{f}]\), we obtain  
\[z_{2,1}(t)=D_{\mathrm{u}2}^{-1}(t)\big{(}-A_{2}^{T}(t)\bar{\lambda}_{\mathrm{ u}1,1}(t)+D_{\mathrm{v}2}(t)\bar{\lambda}_{\mathrm{u},\mathrm{v}1}(t)\big{)}, \quad t\in[0,t_{f}]. \tag{73}\]  
By substitution of (73) into the first and fifth equations of System (70), we obtain (quite similarly to System (39)) the following system of four differential equations with respect to \(\bar{z}_{1,1}(t)\), \(\bar{\lambda}_{\mathrm{u}1,1}(t)\), \(\bar{\lambda}_{\mathrm{v}1}(t)\), and \(\bar{\lambda}_{\mathrm{u},\mathrm{v}1}(t)\) in the time interval \([0,t_{f}]\):  
\[\frac{d\bar{z}_{1,1}(t)}{dt}=A_{1}(t)\bar{z}_{1,1}(t)-A_{2}(t)D_{ \mathrm{u}2}^{-1}(t)A_{2}^{T}(t)\bar{\lambda}_{\mathrm{u}1,1}(t)-S_{\mathrm{v }1}(t)\bar{\lambda}_{\mathrm{v}1}(t)\] \[+A_{2}(t)D_{\mathrm{u}2}^{-1}(t)D_{\mathrm{v}2}(t)\bar{\lambda}_{ \mathrm{u},\mathrm{v}1}(t),\] \[\frac{d\bar{\lambda}_{\mathrm{u}1,1}(t)}{dt}=-D_{\mathrm{u}1}(t )\bar{z}_{1,1}(t)-A_{1}^{T}(t)\bar{\lambda}_{\mathrm{u}1,1}(t)+D_{\mathrm{v}1 }(t)\bar{\lambda}_{\mathrm{u},\mathrm{v}1}(t),\] \[\frac{d\bar{\lambda}_{\mathrm{v}1}(t)}{dt}=-D_{\mathrm{v}1}^{T}(t )\bar{z}_{1,1}(t)+D_{\mathrm{v}2}^{T}(t)D_{\mathrm{u}2}^{-1}(t)A_{2}^{T}(t) \bar{\lambda}_{\mathrm{u}1,1}(t)-A^{T}(t)\bar{\lambda}_{\mathrm{v}1}(t)\] \[-D_{\mathrm{v}2}^{T}(t)D_{\mathrm{u}2}^{-1}(t)D_{\mathrm{v}2}(t) \bar{\lambda}_{\mathrm{u},\mathrm{v}1}(t),\] \[\frac{d\bar{\lambda}_{\mathrm{u},\mathrm{v}1}(t)}{dt}=S_{\mathrm{ v}1}^{T}(t)\bar{\lambda}_{\mathrm{u}1,1}(t)-S_{\mathrm{u},\mathrm{v}}(t)\bar{ \lambda}_{\mathrm{v},1}(t)+A(t)\bar{\lambda}_{\mathrm{u},\mathrm{v}1}(t). \tag{74}\]  
The system of the differential Equations (74) is subject to the boundary conditions (71). Similarly to System (39), System (74) is of the lower dimension than the differential system in (17). **Remark 9**. : _Let us note the following. The boundary-value Problems (74) and (71) are of the same form as the boundary-value Problems (39) and (37). Since, due to Assumption (A7), Problems (39) and (37) have the unique solution, then Problems (74) and (71) also have the unique solution \(\mathrm{col}\big{(}\bar{z}_{1,1}(t),\bar{\lambda}_{\mathrm{u}1,1}(t),\bar{ \lambda}_{\mathrm{v}1}(t),\bar{\lambda}_{\mathrm{u},\mathrm{v}1}(t)\big{)},\ t\in[0,t_{f}]\)._Thus, the solution to the boundary-value Problems (74) and (71), along with \(\bar{\lambda}_{u2,1}(t)\) (see Equation (72)) and \(z_{2,1}(t)\) (see Equation (73)), constitutes the coefficient of the first-order (with respect to \(\varepsilon\)) addend in the outer solution.