Fractional-feedback stabilization for a class of evolution systems

TL;DR

This paper investigates stabilization techniques for evolution systems with fractional damping, demonstrating well-posedness, strong stability, and polynomial stabilization, with applications to wave equations.

Contribution

It introduces a comprehensive approach combining LaSalle's invariance, resolvent, and multiplier techniques for fractional-damped systems, including new polynomial stabilization results.

Findings

System is well posed

Energy is strongly stable

Achieves polynomial stabilization

Abstract

We study the problem of stabilization for a class of evolution systems with fractional-damping. After writing the equations as an augmented system we prove in this article first that the problem is well posed. Second, using the LaSalle's invariance principle we show that the energy of the system is Strongly stable. Then, based on a resolvent approach we show a luck of uniform stabilization. Next, using multiplier techniques combined with the frequency domain method, we shall give a polynomially stabilization result under some consideration on the stabilization of an auxiliary dissipating system. Finally, we give some applications to the wave equation.