## 4 Asymptotic Solution to the Boundary-Value Problem (17)
### First-Order Formal Asymptotic Solution to Problems (26) and (27)
Therefore, in (37), boundary conditions for \(\bar{\lambda}_{\text{u2,0}}(t)\) and \(\bar{z}_{2,0}(t)\) are absent._Resolving the fourth equation in System (36) with respect to \(\bar{z}_{2,0}(t)\), and taking into account the invertibility of the matrix \(D_{\mathrm{u},2}(t)\) for all \(t\in[0,t_{f}]\), we obtain  
\[\bar{z}_{2,0}(t)=D_{\mathrm{u},2}^{-1}(t)\big{(}-A_{2}^{T}(t)\bar{\lambda}_{ \mathrm{u}1,0}(t)+D_{\mathrm{v},2}(t)\bar{\lambda}_{\mathrm{u},\mathrm{v},0}(t )\big{)},\quad t\in[0,t_{f}]. \tag{38}\]  
Substitution of (38) into the first and fifth equations of System (36), use of Equations (20)-(23), and rearrangement of the sixth equation of this system yield after a routine algebra the following set of four differential equations with respect to \(\bar{z}_{1,0}(t)\), \(\bar{\lambda}_{\mathrm{u}1,0}(t)\), \(\bar{\lambda}_{\mathrm{v},0}(t)\), and \(\bar{\lambda}_{\mathrm{u},\mathrm{v},0}(t)\) in the time interval \([0,t_{f}]\):  
\[\frac{d\bar{z}_{1,0}(t)}{dt}=A_{1}(t)\bar{z}_{1,0}(t)-A_{2}(t)D_{ \mathrm{u},2}^{-1}(t)A_{2}^{T}(t)\bar{\lambda}_{\mathrm{u}1,0}(t)-S_{\mathrm{ v}1}(t)\bar{\lambda}_{\mathrm{v},0}(t)\] \[+A_{2}(t)D_{\mathrm{u},2}^{-1}(t)D_{\mathrm{v},2}(t)\bar{\lambda }_{\mathrm{u},\mathrm{v},0}(t),\] \[\frac{d\bar{\lambda}_{\mathrm{u}1,0}(t)}{dt}=-D_{\mathrm{u},1}( t)\bar{z}_{1,0}(t)-A_{1}^{T}(t)\bar{\lambda}_{\mathrm{u}1,0}(t)+D_{\mathrm{v},1}(t) \bar{\lambda}_{\mathrm{u},\mathrm{v},0}(t),\] \[\frac{d\bar{\lambda}_{\mathrm{v},0}(t)}{dt}=-D_{\mathrm{v},1}^{T }(t)\bar{z}_{1,0}(t)+D_{\mathrm{v},2}^{T}(t)D_{\mathrm{u},2}^{-1}(t)A_{2}^{T} (t)\bar{\lambda}_{\mathrm{u}1,0}(t)-A^{T}(t)\bar{\lambda}_{\mathrm{v},0}(t)\] \[-D_{\mathrm{v},2}^{T}(t)D_{\mathrm{u},2}^{-1}(t)D_{\mathrm{v},2} (t)\bar{\lambda}_{\mathrm{u},\mathrm{v},0}(t),\] \[\frac{d\bar{\lambda}_{\mathrm{u},\mathrm{v},0}(t)}{dt}=S_{ \mathrm{v},1}^{T}(t)\bar{\lambda}_{\mathrm{u}1,0}(t)-S_{\mathrm{u},\mathrm{v }}(t)\bar{\lambda}_{\mathrm{v},0}(t)+A(t)\bar{\lambda}_{\mathrm{u},\mathrm{v},0}(t). \tag{39}\]  
The system of differential Equations (39) is subject to boundary Conditions (37). **Remark 7**. : _Let us note the following. The dimension of each of the state variables \(z(t)\), \(\lambda_{\mathrm{u}}(t)\), \(\lambda_{\mathrm{v}}(t)\), and \(\lambda_{\mathrm{u},\mathrm{v}}(t)\) of the differential system in (17) is \(n\). The dimension of the state variables \(\bar{z}_{1,0}(t)\), \(\bar{\lambda}_{\mathrm{u}1,0}(t)\) of the differential system (39) is \(n-r\), while the dimension of the state variables \(\bar{\lambda}_{\mathrm{v},0}(t)\), \(\bar{\lambda}_{\mathrm{u},\mathrm{v},0}(t)\) of this system is \(n\). Hence, differential System (39) is of a lower dimension than differential System (17). Moreover, in contrast with System (17), System (39) is independent of \(\varepsilon\)._  
In what follows, we assume the following. **(A7)** The boundary-value Problems ((39) and (37)) have the unique solution  
\(\mathrm{col}\big{(}\bar{z}_{1,0}(t),\bar{\lambda}_{\mathrm{u}1,0}(t),\bar{ \lambda}_{\mathrm{v}0}(t),\bar{\lambda}_{\mathrm{u},\mathrm{v},0}(t)\big{)}, \quad t\in[0,t_{f}]\). Thus, the solution to the boundary-value Problems ((39) and (37)), along with \(\bar{\lambda}_{\mathrm{u}2,0}(t)=0\) (see the second equation in (36)) and \(\bar{z}_{2,0}(t)\) (see Equation (38)), constitute the zero-order (with respect to \(\varepsilon\)) outer solution.