## ON THE CHARACTERIZATION OF PROPERLY RELAXED DELAYED CONTROLS

### J. Rosenblueth, J. Warga and Q. J. Zhu

Abstract: In previous papers Warga and Zhu have proven the conjecture of Rosenblueth and Vinter that a particular abstract model characterizes the set $\bar{\Re}$ of relaxed controls that are limits of ordinary controls with (possibly time-dependent) delays. Rosenblueth and Vinter have also provided a simple and concrete characterization of $\bar{\Re}$ for the case of constant delays that are consecutive multiples of a common divisor $h$. In the present paper we provide a concrete characterization of $\bar \Re$ for the case of delays that are arbitrary (and not necessarily consecutive) multiples of some $h$. We also exhibit a simple compactification of the set of solutions of controlled differential equations with such delays in the control.