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 \(\varepsilon^{\gamma}(\varepsilon^{k}+\mu^{k})\), \(0<\gamma<1\).

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具有两个小参数的奇摄动时间最优控制问题解的渐近性

该论文继续了作者先前的研究。我们考虑具有两个独立的小参数并且以球\（\ left \ {\ begin {array} [] {llll} \的形式存在于控件上的平滑几何约束的奇摄动线性自治系统的时间最优控制问题。幻影{\ 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。\）其中\（J = \ left（\ begin {array} [] {rr} 0＆1 \\ 0＆0 \ end {array} \ right）。\）这种情况与先前研究的具有快变量和慢变量的系统的主要区别在于，此处快变量处的矩阵 \（J \）是特征值为零的二阶Jordan块，因此不满足标准渐近线稳定条件。继续研究，我们根据第二个小参数\（\ mu \）考虑初始条件 。我们从渐近序列\（\ varepsilon ^ {\ gamma}（\ varepsilon ^ {k} + \ mu ^ {k}）的最佳时间和最优控制的Erdelyi意义上推导并证明一个完整的渐近展开\），\（0 <\ gamma <1 \）。