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

## TL;DR

This paper develops a stabilization method for a class of parabolic integro-differential equations using finite-dimensional feedback control, with applications to viscoelastic fluid models like Oldroyd B and Jeffreys.

## Contribution

It introduces a new stabilization approach for PIDE in Hilbert spaces via Riccati-based feedback, applicable to complex viscoelastic fluid models.

## Key findings

- Achieved exponential decay rate stabilization.
- Derived explicit feedback law from algebraic Riccati equation.
- Validated approach on viscoelastic fluid models.

## Abstract

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