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
Received June 1, 2024
Revised June 13, 2024
Accepted June 17, 2024
Funding Agency: The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2024-1377).
Mikhail Ivanovich Gusev, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: gmii@imm.uran.ru
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Cite this article as: M.I. Gusev. On some properties of reachable sets for nonlinear systems with control constraints in $L_p$. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 3, pp. 99–112.
