Existence, uniqueness, and stabilization results for parabolic variational inequalities

TL;DR

This paper develops feedback stabilization methods for parabolic variational inequalities with obstacle constraints, achieving exponential decay of the state variable using a finite number of actuators, supported by numerical examples.

Contribution

It introduces a novel feedback stabilization approach for parabolic variational inequalities with obstacle constraints, utilizing a Moreau-Yosida approximation and finite actuators.

Findings

Exponential stabilization to nonstationary trajectories achieved.

Feedback operator constructed with a finite number of actuators.

Numerical examples demonstrate effectiveness for smooth and nonsmooth obstacles.

Abstract

In this paper we consider feedback stabilization for parabolic variational inequalities of obstacle type with time and space depending reaction and convection coefficients and show exponential stabilization to nonstationary trajectories. Based on a Moreau-Yosida approximation, a feedback operator is established using a finite (and uniform in the approximation index) number of actuators leading to exponential decay of given rate of the state variable. Several numerical examples are presented addressing smooth and nonsmooth obstacle functions.