  There exist many ways to stabilize an infinite-dimensional linear autonomous
control systems when it is possible. Anyway, finding an exponentially
stabilizing feedback control that is as simple as possible may be a challenge.
The Riccati theory provides a nice feedback control but may be computationally
demanding when considering a discretization scheme. Proper Orthogonal
Decomposition (POD) offers a popular way to reduce large-dimensional systems.
In the present paper, we establish that, under appropriate spectral
assumptions, an exponentially stabilizing feedback Riccati control designed
from a POD finite-dimensional approximation of the system stabilizes as well
the infinite-dimensional control system.
