Asymptotics of the Solution to a Singularly Perturbed Time-Optimal Control Problem with Two Small Parameters

The paper continues the authors' previous studies. We consider a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball
$$\left\{ \begin{array}{llll} \phantom{\varepsilon^3}\dot{x}=y,\,& x,\,y\in \mathbb{R}^{2},\quad u\in \mathbb{R}^{2},\\[1ex] \varepsilon^3\dot{y}=Jy+u,&\,\|u\|\le 1,\quad 0<\varepsilon,\mu\ll 1,\\[1ex] x(0)=x_0(\varepsilon,\mu)=(x_{0,1}, \varepsilon^3\mu\xi)^*,\quad y(0)=y_0,\\[1ex] x(T_{(\varepsilon,\mu})=0,\quad y(T_{(\varepsilon,\mu})=0,\quad T_{(\varepsilon,\mu} \to \min,& \end{array} \right.$$
where \vspace{-1mm}
$$J=\left( \begin{array}{rr} 0&1 \\ 0&0\end{array} \right).$$
The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix $J$ at the fast variables is the second-order Jordan block with zero eigenvalue and, thus, does not satisfy the standard asymptotic stability condition. Continuing the research, we consider initial conditions depending on the second small parameter $\mu$. We derive and justify a complete asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to the asymptotic sequence $\varepsilon^\gamma(\varepsilon^k+\mu^k)$, $0<\gamma<1$.