Decay of Hankel singular values of analytic control systems

We show that control systems with an analytic semigroup and control and observation operators that are not too unbounded have a Hankel operator that belongs to the Schatten class Sp for all positive p. This implies that the Hankel singular values converge to zero faster than any polynomial rate. This in turn implies fast convergence of balanced truncations. As a corollary, decay rates for the eigenvalues of the controllability and observability gramians are also provided. Applications to the heat equation and a plate equation are given.