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 SpSp for all positive pp. 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.

Research highlights
► The Hankel operator of an analytic control system belongs to every Schatten class.
► The Hankel singular values of an analytic control system decay rapidly.
► The eigenvalues of the gramians of an analytic control system decay rapidly.
► Analytic control systems can be approximated very well by balanced truncation.