Nuclearity of Hankel operators for ultradifferentiable control systems

Nuclearity of the Hankel operator is a known sufficient condition for convergence of Lyapunov-balanced truncations. We show how a previous result on nuclearity of Hankel operators of systems with an analytic semigroup can be extended to systems with a semigroup of class DpDp with p≥1p≥1 (the case p=1p=1 being the analytic case). For semigroups that are generated by a Dunford–Schwartz spectral operator we prove that being of class DpDp is equivalent to being (Gevrey) ultradifferentiable of order pp. We illustrate how for certain partial differential equations our results lead to an easy way of showing nuclearity of the Hankel operator for a wide range of control and observation operators by considering several examples of damped beams.