Uniform exponential stabilization of distributed bilinear parabolic time delay systems with bounded feedback control

In this paper we deal with the problem of uniform exponential stabilization for a class of distributed bilinear parabolic systems with time delay in a Hilbert space by means of a bounded feedback control. The uniform exponential stabilization problem of such a system reduces to stabilizing only its projection on a suitable finite dimensional subspace. Furthermore, the stabilizing feedback control depends only on the state projection on the finite dimensional subspace. An explicit decay rate estimate of the stabilized state is given provided that a nonstandard weaker observability condition is satisfied. Illustrative examples for partial functional differential equations are displayed.