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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2024 Volume 30, Number 3, Pages 99–112 (Mi timm2107)

On some properties of reachable sets for nonlinear systems with control constraints in $L_p$

M. I. Gusev

N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: The paper considers reachable sets at a given time for control-affine systems with integral control constraints in the space $L_p$ for $p>1$. The goal of the paper is to characterize controls leading to the boundary of reachable sets as solutions to extremal problems and to study the necessary optimality conditions in the form of the Pontryagin maximum principle for these controls. A reachable set is interpreted here as the image of the set of admissible controls under a nonlinear mapping defined by a dynamical system. We also study the application of the maximum principle to describe projections of a reachable set onto a subspace and its sections by a hyperplane. The dependence of a reachable set on the control resource is studied. The results obtained are illustrated using the example of linear systems. It is shown that in this case the optimality conditions for boundary controls are necessary and sufficient.

Keywords: control system, integral constraints, reachable set, nonlinear mapping, maximum principle.

UDC: 517.977

MSC: 93B03, 49K15

Received: 01.06.2024
Revised: 13.06.2024
Accepted: 17.06.2024

DOI: 10.21538/0134-4889-2024-30-3-99-112


 English version:
Proceedings of the Steklov Institute of Mathematics (Supplement Issues), 2024, 327, suppl. 1, S124–S137

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© Steklov Math. Inst. of RAS, 2025