## 5 Main Results
### Asymptotically Suboptimal Stackelberg Solution to the Game (9)-(11)  
Consider the following controls for the leader and the follower, respectively,  
\[\widetilde{u}(t,\varepsilon)=-\frac{1}{\varepsilon}\lambda_{u2,0}^{0}(t/ \varepsilon)-\bar{\lambda}_{u2,1}(t),\quad t\in[0,t_{f}],\quad\varepsilon\in( 0,\varepsilon_{0}]\]  
and  
\[\widetilde{v}(t)=-B_{v}^{T}(t)\bar{\lambda}_{v,0}(t),\quad t\in[0,t_{f}]. \tag{117}\]  
Let \(\widetilde{J}_{u}(\varepsilon)\) and \(\widetilde{J}_{v}(\varepsilon)\) be the values of the functionals \(J_{u}(u,v)\) and \(J_{v}(u,v)\), respectively, generated by the pair of the controls \(\big{(}\widetilde{u}(t,\varepsilon),\widetilde{v}(t)\big{)}\) in the game (9)-(11). **Theorem 3**: _Let Assumptions (A1)-(A7) be valid. Then, for all \(\varepsilon\in(0,\varepsilon_{0}]\), the following inequalities are satisfied:_  
\[\big{|}J_{u}^{*}(\varepsilon)-\widetilde{J}_{u}(\varepsilon)\big{|} \leq\widetilde{b}\varepsilon,\] \[\big{|}J_{v}^{*}(\varepsilon)-\widetilde{J}_{v}(\varepsilon) \big{|} \leq\widetilde{b}\varepsilon, \tag{118}\]  
_where \(\widetilde{b}>0\) is some constant independent of \(\varepsilon\)._  
Proof. : Let us start with the proof of the first inequality in (118). Substituting the open-loop Stackelberg solution \(\big{(}u^{*}(t,\varepsilon),v^{*}(t,\varepsilon)\big{)}\) into System (9) instead of \(\big{(}u(t),v(t)\big{)}\), we obtain, for all \(\varepsilon\in(0,\varepsilon_{0}]\),  
\[\frac{dz(t)}{dt}=A(t)z(t)+B_{u}(t)u^{*}(t,\varepsilon)+B_{v}(t)v^{*}(t, \varepsilon),\quad t\in[0,t_{f}],\ \ z(0)=z_{0}. \tag{119}\]  
The solution \(z^{*}(t,\varepsilon)\) of this system is the Stackelberg optimal trajectory of the game (9)-(11). Due to Proposition 2 and the equivalence of the boundary-value Problems (17), (26), and (27), we directly have  
\[z^{*}(t,\varepsilon)=z(t,\varepsilon),\quad t\in[0,t_{f}],\quad\varepsilon\in (0,\varepsilon_{0}], \tag{120}\]  
where \(z(t,\varepsilon)\) is the corresponding component of the solution to the boundary-value Problems (26) and (27). Furthermore, by virtue of the equation for \(\lambda_{\rm v}^{1}(t,\varepsilon)\) and the inequality for \(\lambda_{\rm v}(t,\varepsilon)\) (see Equations (102) and (103)), we have the inequality  
\[\big{\|}\lambda_{\rm v}(t,\varepsilon)-\bar{\lambda}_{{\rm v},0}(t)\big{\|} \leq c_{4}\varepsilon,\quad t\in[0,t_{f}],\quad\varepsilon\in(0,\varepsilon_{ 0}], \tag{129}\]  
where \(c_{4}>0\) is some constant independent of \(\varepsilon\). Now, Equations (116), (122), and (123), and Inequalities (125), (128), and (129), yield the first inequality in (118). The second inequality in (118) is proven similarly. This completes the proof of the theorem. **Remark 11**. : _Due to Theorem 3 and Equation (16), the pair of the controls \(\big{(}\widetilde{u}(t,\varepsilon),\widetilde{v}(t)\big{)}\) is an asymptotically suboptimal Stackelberg solution in the game (9)-(11). To obtain this solution, one has to first solve the boundary-value Problems (39) and (37). Then, using the explicit formulas in (38), (60), and (72), one calculates \(\lambda_{{\rm u}2,0}^{0}(t/\varepsilon)\) and \(\bar{\lambda}_{{\rm u}2,1}(t)\)._  
As a direct consequence of Theorems 2 and 3, we obtain the following assertion. **Corollary 1**. : _Let Assumptions (A1)-(A7) be valid. Then, for all \(\varepsilon\in(0,\varepsilon_{0}]\), the values \(\widetilde{J}_{\rm u}(\varepsilon)\) and \(\widetilde{J}_{\rm v}(\varepsilon)\) satisfy the inequalities_  
\[\big{|}\widetilde{J}_{\rm u}(\varepsilon)-\widetilde{J}_{{\rm u},0}\big{|} \leq(b+\widetilde{b})\varepsilon,\] \[\big{|}\widetilde{J}_{\rm v}(\varepsilon)-\widetilde{J}_{{\rm v},0}\big{|} \leq(b+\widetilde{b})\varepsilon. \tag{130}\]  
**6. Example**  
Consider a particular case of the Stackelberg game (9)-(11) with the following data:  
\[n=2,\quad r=1,\quad s=1,\quad t_{f}=1,\] \[A(t)=A=\left( \begin{array}{cc}0&0\\ 0&0\end{array} \right)\!\!\!,\quad B_{\rm u}(t)=B_{\rm u}=\left( \begin{array}[] {cc}0\\ 1\end{array} \right)\!\!\!,\quad B_{\rm v}(t)=B_{\rm v}=\left( \begin{array}[] {cc}1\\ -1\end{array}