The present work is devoted to a time-optimal control problem for a singularly perturbed linear autonomous system with smooth geometric constraints on the control and an unbounded target set:
$$\left\{ \begin{array}{ll} \phantom{\varepsilon}\dot{x}= A_{11}x + A_{12}y + B_1 u, & x\in \mathbb{R}^{n},\,y\in \mathbb{R}^{m},\,u\in\mathbb{R}^{r},\\[1ex]
\varepsilon\dot{y}=A_{21}x + A_{22}y + B_2 u,& \|u\|\le 1,\\[1ex] x(0)=x_0\not=0,\quad y(0)=y_0, & 0<\varepsilon\ll 1,\\[1ex]
x(T_\varepsilon)=0,\quad y(T_\varepsilon)\in \mathbb{R}^{m},\quad T_\varepsilon \longrightarrow \min. \end{array} \right. $$
The uniqueness of the representation of the optimal control with a normalized defining vector in the limit problem is proved. The solvability of the problem is established. The limit relations for the optimal time and the vector determining the optimal control are obtained. An asymptotic analog of the implicit function theorem is proved and used to derive a complete asymptotics of the solution to the problem in powers of the small parameter $\varepsilon$.
Keywords: optimal control, time-optimal control problem, asymptotic expansion, singularly perturbed problem, small parameter
Received January 15, 2020
Revised February 27, 2020
Accepted March 2, 2020
Funding Agency: O.O. Kovrizhnykh’s research is supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).
Aleksei Rufimovich Danilin, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: dar@imm.uran.ru
Ol’ga Olegovna Kovrizhnykh, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: koo@imm.uran.ru
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Cite this article as: A.R. Danilin, O.O. Kovrizhnykh. Asymptotics of a solution to a singularly perturbed time-optimal control problem of transferring an object to a set. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 132–146; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2020, Vol. 313, Suppl. 1, pp. S40–S53.
