Observation of Volterra systems with scalar kernels (Q2427788)

This paper is devoted to a study of the abstract observed Volterra equation \[ \begin{align*}{ x(t) &= a(t) - \int_0^t a(t-s) A x(s) ds,\qquad t \ge 0,\cr y(t) &= C x(t).\cr }\end{align*} \] Here \(A\) is a closed operator with dense domain in a Banach space \(X\), and \(a\) is a scalar kernel which is of sub-exponential growth, \textit{\(1\)-regular}, and \textit{parabolic} in the sense that its Laplace transform \(\hat a\) satisfies \(\hat a(\lambda) \neq 0\), \(1/\hat a(\lambda) \in \rho(A)\), \((1 - \hat a(\lambda)A)^{-1} \leq K\), and \(|\lambda \hat a '(\lambda)| \leq K |\hat a(\lambda)|\) for some finite constant \(K\) and all \(\Re \lambda > 0\). The operator \(C\) is an operator from \(X\) into another Banach space \(Y\) whose restriction to the domain of \(A\) is bounded with respect to the graph norm of \(A\). Two different sets of sufficient conditions are given for the operator \(C\) to be finite-time admissible, which means that there exist constants \(\eta\), \(K > 0\) such that \[ \left( \int_0^t || CS(r) x||^2 dr\right) ^{1/2} \leq K e^{\eta t} || x ||, \] for all \(t \geq 0\) and all \(x\) in the domain of \(A\); here \(S\) is the so-called \textit{solution family} (or \textit{Volterra resolvent}) of the system. The Laplace transform of \(S\) is denoted by \(H\), and it is given by \[ H(\lambda) = {1 \over \lambda}\, (I - \hat a(\lambda)A)^{-1},\qquad \Re \lambda > 0. \] One of the sufficient conditions for the finite-time admissibility of \(C\) is that \[ \sup_{r > 0} \left\| (1 + \log^+ r)^\alpha r^{1/2} CH(r) \right\| < \infty \] for some \(\alpha > 1/2\). The above condition is known to be \textit{necessary} for \(\alpha = 0\). The second sufficient condition requires \(A\) to generate an exponentially stable semigroup, and it is based on a subordination argument to obtain admissibility of \(C\) for the observed Volterra equation from the admissibility of \(C\) for the underlying Cauchy problem.