On The Characterization Of Properly Relaxed Delayed Controls

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In previous papers Warga and Zhu have proven the conjecture of Rosenblueth and Vinter that a particular abstract model characterizes the set R 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 R 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 R 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 control.