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

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.