The paper considers reachable sets at a given time of linear control systems with integral constraints on control in the form of a ball in the space $L_p$ for $p>1$. The reachable sets are convex compact sets in the finite-dimensional Euclidean space $\mathbb R^n$. For $p=2$, it is known that the reachable set, under the controllability condition, is an ellipsoid in $\mathbb R^n$ whose boundary is a compact smooth manifold diffeomorphic to a sphere. In this paper we obtain sufficient conditions under which the boundary of the attainability set turns out to be a smooth manifold of dimension $n-1$ for all $1<p\leq 2$.
Keywords: control system, integral constraints, reachable set, nonlinear mapping, maximum principle
Received March 04, 2025
Revised March 27, 2025
Accepted April 1, 2025
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 075-02-2025-1549).
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: gmi@imm.uran.ru
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Cite this article as: M.I. Gusev. On smoothness of the boundary of a reachable sets under integral control constraints. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol 31, no. 2, pp. 81–93.
