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 systems in the case where the operator of the associated Cauchy problem generates a semigroup on a Banach space, and the integral part is given by a convolution with an $L^p$-admissible observation operators kernel with $p\in [1,\infty )$. Sufficient and/or necessary conditions for $L^p$-admissibility of control and observation operators are given in term of kernels under which $L^p$-admissibility for Volterra integro-differential system follows from that of the corresponding Cauchy system without convolution term. In particular, the 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. Our results are generalization of those known to hold for standard Cauchy problems. Particular attention is paid to the problem of obtaining the input–output representation of such systems, providing a theory which is analogous to Salamon–Weiss for linear systems. We mention that our approach is mainly based on the theory of infinite-dimensional $L^p$-well-posed 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.