On a singularly perturbed time-optimal control problem with two small parameters
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 2, pp. 76-92
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In this paper we investigate 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. The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix at the fast variables is a multidimensional analog of the second-order Jordan cell with zero eigenvalue and, thus, does not satisfy the standard condition of asymptotic stability. Continuing the research, we consider initial conditions depending on the second small parameter; in the degenerate case, this resulted in an asymptotic expansion of the solution of a fundamentally different type. The solvability of the problem is proved. We also derive and justify a complete power asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to a small parameter at the derivatives in the equations of the systems.
Keywords:
optimal control, time-optimal control problem, asymptotic expansion, singularly perturbed problem, small parameter.
@article{TIMM_2018_24_2_a8,
author = {A. R. Danilin and O. O. Kovrizhnykh},
title = {On a singularly perturbed time-optimal control problem with two small parameters},
journal = {Trudy Instituta matematiki i mehaniki},
pages = {76--92},
publisher = {mathdoc},
volume = {24},
number = {2},
year = {2018},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TIMM_2018_24_2_a8/}
}
TY - JOUR AU - A. R. Danilin AU - O. O. Kovrizhnykh TI - On a singularly perturbed time-optimal control problem with two small parameters JO - Trudy Instituta matematiki i mehaniki PY - 2018 SP - 76 EP - 92 VL - 24 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TIMM_2018_24_2_a8/ LA - ru ID - TIMM_2018_24_2_a8 ER -
%0 Journal Article %A A. R. Danilin %A O. O. Kovrizhnykh %T On a singularly perturbed time-optimal control problem with two small parameters %J Trudy Instituta matematiki i mehaniki %D 2018 %P 76-92 %V 24 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TIMM_2018_24_2_a8/ %G ru %F TIMM_2018_24_2_a8
A. R. Danilin; O. O. Kovrizhnykh. On a singularly perturbed time-optimal control problem with two small parameters. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 2, pp. 76-92. http://geodesic.mathdoc.fr/item/TIMM_2018_24_2_a8/