ASYMPTOTIC BEHAVIOR OF REACHABLE SETS WITH \(L_p\)-BOUNDED CONTROLS

Mikhail I. Gusev     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)

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

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References


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DOI: http://dx.doi.org/10.15826/umj.2025.2.009

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