Asymptotic expansion of solution to singularly perturbed optimal control problem with convex integral quality functional with terminal part depending on slow and fast variables

We consider an optimal control problem with a convex integral quality functional for a linear system with fast and slow variables in the class of piecewise continuous controls with smooth constraints on the control
$$ \left\{ \begin{aligned} & \dot{x}_{\varepsilon} = A_{11}x_{\varepsilon} + A_{12}y_{\varepsilon}+B_{1}u,\qquad t\in[0,T],\qquad \|u\|\leqslant 1,\\ &\varepsilon\dot{y}_{\varepsilon} = A_{22}y_{\varepsilon} + B_{2}u,\quad x_{\varepsilon}(0)=x^{0},\qquad y_{\varepsilon}(0)=y^{0},\qquad \nabla\varphi_2(0)=0, \\ &J(u)\mathop{:=}\nolimits \varphi_1\left(x_\varepsilon(T)\right) + \varphi_2\left(y_\varepsilon(T)\right) + \int\limits_{0}^{T}\|u(t)\|^2\,dt\rightarrow \min, \end{aligned} \right. $$
where $x\in\mathbb{R}^{n}$, $y\in\mathbb{R}^{m}$, $ u\in\mathbb{R}^{r}$; $A_{ij}$ and $B_{i}$, $i,j=1,2$, are constant matrices of corresponding dimension, and the functions $\varphi_{1}(\cdot), \varphi_{2}(\cdot)$ are continuously differentiable in $\mathbb{R}^{n}, \mathbb{R}^{m},$ strictly convex, and cofinite in the sense of the convex analysis. In the general case, for such problem, the Pontryagin maximum principle is a necessary and sufficient optimality condition and there exist unique vectors $l_\varepsilon$ and $\rho_\varepsilon$ determining an optimal control by the formula
$$ u_{\varepsilon}(T-t):= \frac{C_{1,\varepsilon}^{*}(t)l_\varepsilon + C_{2,\varepsilon}^{*}(t)\rho_\varepsilon} {S\left(\|C_{1,\varepsilon}^{*}(t)l_\varepsilon + C_{2,\varepsilon}^{*}(t)\rho_\varepsilon\|\right)}, $$
where
\begin{align*} & C_{1,\varepsilon}^{*}(t):= B^*_1 e^{A^*_{11}t} + \varepsilon^{-1}B^*_2\mathcal{W^*}_\varepsilon(t),\quad C_{2,\varepsilon}^{*}(t):= \varepsilon^{-1} B^*_2 e^{A^*_{22} t/\varepsilon}, \\ & \mathcal{W}_\varepsilon(t):= e^{A_{11}t}\int\limits_{0}^{t} e^{-A_{11}\tau}A_{12}e^{A_{22} \tau/\varepsilon}\,d\tau, \quad S(\xi)\mathop{:=}\nolimits \left\{ \begin{aligned} & 2,\qquad 0\leqslant \xi\leqslant2, \\ &\xi, \qquad \xi>2. \end{aligned} \right. \end{align*}
The main difference of our problem from the previous papers is that the terminal part of quality functional depends on the slow and fast variables and the controlled system is a more general form. We prove that in the case of a finite number of control change points, a power asymptotic expansion can be constructed for the initial vector of dual state $\lambda_\varepsilon=\left(l_\varepsilon^*\: \rho_\varepsilon^*\right)^*$, which determines the type of the optimal control.