Summary: In contrast with \({\mathcal H}_\infty\) and \({\mathcal H}_2\) control theories, the problem of persistent disturbance rejection (\(l^1\) optimal control) leads to dynamic controllers, even when the states of the plant are available for feedback. Using viability theory, it has recently been shown, in a nonconstructive way [\textit{J. S. Shamma}, Syst. Control Lett. 21, 265-270 (1993; Zbl 0798.93030)], that in the state- feedback case the same performance achieved by any dynamic linear time- invariant controller can be achieved using memoryless nonlinear state feedback. Here, we give an alternative, constructive proof of these results for discrete- and continuous-time systems. The main result of the paper shows that in both cases, the \(l^1\) norm achieved by any stabilizing state-feedback linear dynamic controller can also be achieved using a memoryless variable structure controller.