Admissibility and Controllability of diagonal Volterra equations with scalar inputs

TL;DR

This paper investigates the control properties of diagonal Volterra equations with scalar inputs, establishing conditions for admissibility and controllability through advanced functional analysis and illustrating with heat conduction examples.

Contribution

It introduces new criteria for admissibility and controllability of Volterra equations using Carleson measures and interpolation theory, extending known results to systems with memory effects.

Findings

Conditions for admissibility via Carleson measures

Characterization of controllability through interpolation of analytic functions

Application to heat conduction with memory

Abstract

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.