Abstract:
In this paper, we consider the problem of constructing external estimates of reachable sets as a level set of a certain differentiable Lyapunov–Bellman function (depending only on the state vector) for a control system with an integral control constraint. In particular, with its suitable choice, one can obtain ellipsoidal and rectangular estimates. The proposed constructions are based on integral estimates, the maximum solution, and the comparison principle for systems of differential inequalities. By using time in the arguments of the Lyapunov–Bellman function, it is possible to obtain more accurate estimates. In the linear nonstationary case, the latter can coincide with the set of reachability. A number of illustrative examples for nonlinear systems are given.
Keywords:
reachable set, controlled system, integral constraints, integral inequalities, comparison principle, external estimates.
\Bibitem{Zyk19}
\by I.~V.~Zykov
\paper On external estimates of reachable sets of control systems with integral constraints
\jour Izv. IMI UdGU
\yr 2019
\vol 53
\pages 61--72
\mathnet{http://mi.mathnet.ru/iimi371}
\crossref{https://doi.org/10.20537/2226-3594-2019-53-06}
\elib{https://elibrary.ru/item.asp?id=38503199}
Linking options:
https://www.mathnet.ru/eng/iimi371
https://www.mathnet.ru/eng/iimi/v53/p61
This publication is cited in the following 4 articles:
M. S. Nikol'skii, “On the continuity of the optimal time as a function of the initial state for linear controlled objects with integral constraints on controls”, Proc. Steklov Inst. Math. (Suppl.), 325, suppl. 1 (2024), S147–S154
M. I. Gusev, I. O. Osipov, “O zadache lokalnogo sinteza dlya nelineinykh sistem s integralnymi ogranicheniyami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 32:2 (2022), 171–186
I. V. Zykov, “Priblizhennoe vychislenie mnozhestv dostizhimosti lineinykh upravlyaemykh sistem pri raznotipnykh ogranicheniyakh na upravlenie”, Izv. IMI UdGU, 60 (2022), 16–33
M. I. Gusev, I. O. Osipov, “Asymptotic Behavior of Reachable Sets on Small Time Intervals”, Proc. Steklov Inst. Math. (Suppl.), 309, suppl. 1 (2020), S52–S64
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