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In this paper, we prove the stabilizability of abstract ParabolicIntegro-Differential Equations (PIDE) in a Hilbert space with decay rate$e^{-\gamma t} $ for certain $\gamma > 0,$ by means of a finite dimensionalcontroller in the feedback form. We determine a linear feedback law which isobtained by solving an algebraic Riccati equation. To prove the existence ofthe Riccati operator, we consider a linear quadratic optimal control problemwith unbounded observation operator.  The abstract theory of stabilization developed here is applied to specificproblems related to viscoelastic fluids, e.g. Oldroyd B model and Jeffreysmodel.

Internal Stabilization of a Class of Parabolic Integro-Differential Equations: Application to Viscoelastic Fluids