Asymptotic expansion of a solution for one singularly perturbed optimal control problem with a convex integral quality index depends on slow variables and smooth control constraints

The paper deals with the problem of optimal control with a convex integral quality index depends on slow variables for a linear steady-state control system with a fast and slow variables in the class of piecewise continuous controls with a smooth control constraints
$$\begin{cases} \dot{x}_{\varepsilon} = A_{11}x_{\varepsilon}+A_{12}y_{\varepsilon}+B_{1}u, & t\in[0,T], \qquad \|u\|\leqslant 1,\\ \varepsilon\dot{y}_{\varepsilon} = A_{21}x_{\varepsilon}+A_{22}y_{\varepsilon}+B_{2}u, & x_{\varepsilon}(0)=x^{0}, \quad y_{\varepsilon}(0)=y^{0},\\ J_\varepsilon(u):= \varphi(f(x_{\varepsilon}(T)) + \int_0^T \|u(t)\|^2\,dt\rightarrow \min, \end{cases}$$
where $x_\varepsilon\in\mathbb{R}^{n}$, $y_\varepsilon\in\mathbb{R}^{m}$, $u\in\mathbb{R}^{r}$; $A_{ij}$, $B_{i}$, $i,j=1,2$ — are constant matrices of the corresponding dimension, and $\varphi(\cdot)$ – is the strictly convex and cofinite function that is continuously differentiable in $\mathbb{R}^{n}$ in the sense of convex analysis. In the general case, Pontryagin's maximum principle is a necessary and sufficient optimum condition for the optimization of a such a problem. The initial vector of the conjugate state $l_\varepsilon$ is the unique vector, thus determining the optimal control. It is proven that in the case of a finite number of control switching points, the asymptotics of the vector $l_\varepsilon$ has the character of a power series.