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

Abstract

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 \(\left\{\begin{array}[]{llll}\phantom{\varepsilon^{3}}\dot{x}=y,&x,\,y\in \mathbb{R}^{2},\quad u\in\mathbb{R}^{2},\\ \varepsilon^{3}\dot{y}=Jy+u,&\,\|u\|\leq 1,\quad 0<\varepsilon,\mu\ll 1,\\ x(0)=x_{0}(\varepsilon,\mu)=(x_{0,1},\varepsilon^{3}\mu\xi)^{*},\quad y(0)=y_{ 0},\\ x(T_{\varepsilon,\mu})=0,\quad y(T_{\varepsilon,\mu})=0,\quad T_{\varepsilon, \mu}\to\min,&\end{array}\right.\) where \(J=\left(\begin{array}[]{rr}0&1\\ 0&0\end{array}\right).\) 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 \(\varepsilon^{\gamma}(\varepsilon^{k}+\mu^{k})\), \(0<\gamma<1\).