The paper studies reachable sets at a fixed time instant for nonlinear control systems with integral control constraints given by a ball in the space $L_p$ for $p\geq 1$. It investigates the possibility of approximating these sets by reachable sets under the additional constraint $|u|_{\infty}\leq\nu$ on the control norm in $L_{\infty}$ as $\nu\to\infty$. Sufficient conditions are obtained whose fulfillment guarantees coincidence of the reachable sets for some finite $\nu$ if $p>1$. For $p = 1$, in the linear case error estimates for the approximation in the Hausdorff metric are obtained depending on the magnitude of $\nu$ . Necessary optimality conditions in the form of Pontryagin’s maximum principle are proved for controls that steer the system to the boundary of the reachable set under the double constraints in~$L_1$ and $L_{\infty}$.
Keywords: control system, reachable set, double constraints, asymptotics
Received May 14, 2026
Revised May 27, 2026
Accepted June 1, 2026
Funding Agency: This work was supported by Russian Science Foundation, project No. 25-11-00269, https://rscf.ru/project/25-11-00269/.
Mikhail Ivanovich Gusev, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: gmii@imm.uran.ru
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Cite this article as: M.I. Gusev. Asymptotics of reachable sets under double control constraints. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol 32, no. 3, pp. 57–70.
