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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2022 Volume 213, Number 5, Pages 30–49 (Mi sm9627)

This article is cited in 5 papers

Strong convexity of reachable sets of linear systems

M. V. Balashov

V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

Abstract: The reachable set on some time interval of a linear control system $x'\in Ax\,{+}\,U$, $x(0)=0$, is considered. A number of cases is examined when the reachable set is the intersection of some balls of fixed radius $R$ (that is, a strongly convex set of radius $R$). In some cases the radius $R$ is estimated from above. It turns out that strong convexity is fairly typical for this class of reachable sets in a certain sense.
Among possible applications of this result are the possibility of constructing outer polyhedral approximation of reachable sets with better accuracy in the Hausdorff metric than in the general case, and applications to linear differential games and some optimization problems.
Bibliography: 23 titles.

Keywords: strongly convex set, reachable set, linear control system, Aumann integral, Hausdorff metric, nonsmooth analysis.

MSC: Primary 49J53, 52A20; Secondary 93C05, 90C26

Received: 18.06.2021 and 01.11.2021

DOI: 10.4213/sm9627


 English version:
Sbornik: Mathematics, 2022, 213:5, 604–623

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© Steklov Math. Inst. of RAS, 2025