Necessity of exponential decay for feedback stabilizability

Determine whether uniform exponential decay of the input-coefficient sequence $(|b_n|)$ and the normalized growth sequence $(a_n/|b_n|)$ is necessary for feedback stabilizability of a discrete linear ensemble system, rather than only for pole placement and feedback diagonalizability.

Background

The paper studies feedback stabilization of a countably infinite discrete linear ensemble system consisting of scalar systems with coefficients $(a_n,b_n)$ and a common full-state feedback input. Because spectral placement in the open left half-plane does not by itself guarantee stability for infinite-dimensional systems, the authors analyze pole placement together with diagonalizability of the closed-loop operator.

The paper proves that uniform exponential decay of $(|b_n|)$ is necessary for feedback diagonalizability, and that uniform exponential decay of $(a_n/|b_n|)$ is also necessary under an additional assumption controlling how rapidly $(a_n)$ decays. The sufficient results establish stabilization through the stronger combination of pole placement and feedback diagonalizability. The unresolved issue is whether these decay requirements remain necessary when one asks only for feedback stabilizability, without requiring pole placement or diagonalizability.