On the admissibility and input–output representation for a class of Volterra integro-differential systems

Abstract

This article studies a class of controlled-observed Volterra integro-differential problems in the case that the operator of the associated Cauchy problem generate a semigroup on a Banach space and the integral part being given by a convolution of integrable functions with L p -admissible observation operators kernel. Sufficient and/or necessary conditions for L p -admissibility of control and observation operators are given in term of the kernels. In particular, results on the equivalence between the finite-time (or infinite-time) L p -admissibility and the uniform L p -admissibility are given for both control and observation operators, extending a generalization of results known to hold for the standard Cauchy problems. Particular attention is paid to the problem of obtaining the input-output representation of such systems, providing a theory which is analogue to that of Salamon-Weiss for linear systems. We mention that our approach is mainly based on the theory of infinitedimensional L p -regular linear systems in the Salamon-Weiss sense. These results are illustrated by an example involving heat conduction with memory given by some space fractional Laplacian kernel. KeywordsC 0 -semigroups • Volterra integro-differential problems • Resolvent operators • Admissibility • Weiss conjecture • Regular linear systems