ASYMPTOTIC BEHAVIOR OF REACHABLE SETS WITH $L_p$-BOUNDED CONTROLS

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