Decay of singular values for infinite-dimensional systems with Gevrey regularity

Abstract

We consider the decay rate of the singular values of the input map, the output map and the Hankel operator for a class of infinite-dimensional systems. This class is characterized by the control operator (or the observation operator) having a smoothing effect. We capture this in the definition of a Gevrey operator (which generalizes the known concept of a Gevrey vector). In applications to PDEs, this abstract assumption on the control operator is typically satisfied when the input is multiplied by a function which is a compactly supported Gevrey function in the spatial variable. Using the theory of polynomial approximation (in particular: truncated Chebyshev expansions), we obtain that the singular values decay exponentially in a root of the approximation dimension. The power of the root depends on the order of the Gevrey operator and on whether the underlying semigroup is nilpotent, exponentially stable or polynomially stable.