This article studies Volterra evolution equations from the point of view of control theory, in the case that the generator of the underlying semigroup has a Riesz basis of eigenvectors. Conditions for admissibility of the system's control operator are given in terms of the Carleson embedding properties of certain discrete measures. Moreover, exact and null controllability are expressed in terms of a new interpolation question for analytic functions, providing a generalization of results known to hold for the standard Cauchy problem. The results are illustrated by examples involving heat conduction with memory
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