Asymptotic behavior of reachable sets with $L_p$-bounded controls

Mikhail I. Gusev

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

Abstract: The paper studies the reachable sets of control systems over a fixed time interval, subject to control constraints defined as a ball in the $L_p$ space for $p \geq 1$. The dependence of reachable sets on the parameter $p$ is investigated. For affine-control nonlinear systems, it is established that these sets are continuous in the Hausdorff metric for all $p$, including $p=1$ and $p=\infty$. In the case of linear systems, estimates for the Hausdorff distance between the sets are derived, and their asymptotic behavior as $p\to 1$ and $p\to \infty$ is analyzed. For $p = 1$, the reachable set, up to closure, coincides with the reachable set of the system with impulse control under a constraint on the magnitude of the impulse. The case $p = \infty$ corresponds to geometric (instantaneous) constraints on the control.

Keywords: Reachable set, Control system, Hausdorff continuity, Asymptotics

Language: English

DOI: 10.15826/umj.2025.2.009

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