  In this paper, we investigate the stability of five models of systems. In all
the five models the damping considered is fractional Kelvin-Voigt damping. In
the first model, we consider a Euler-Bernoulli beam and a wave equations
coupled via boundary connections. The localized non-smooth fractional
Kelvin-Voigt damping acts through wave equation only, its effect is transmitted
to the other equation through the coupling by boundary connections. In this
model, we reformulate the system into an augmented model and using a general
criteria of Arendt-Batty, we show that the system is strongly stable. By using
frequency domain approach, combined with multiplier technique we prove that the
energy decays polynomially with rate depends on the order of the fractional
derivative. For the second model, we consider two wave equations coupled
through boundary connections with localized non-regular fractional Kelvin-Voigt
damping acting on one of the two equations. For the third model, we consider
coupled Euler-Bernoulli beam and wave equations through boundary connections
with the same damping, the dissipation acts through the beam equation. We prove
using the same technique as for the first model combined with some
interpolation inequalities and by solving ordinary differential equations of
order 4, that the energy has a polynomial decay rate that depends in the order
of the fractional derivative. In the fourth model, we consider an
Euler-Bernoulli beam with a localized non-regular fractional Kelvin-Voigt
damping. Finally, in the fifth model, we study the polynomial stability of two
Euler-Bernoulli beam equations coupled through boundary connection with a
localized non-regular fractional Kelvin-Voigt damping acting on one of the two
equations. In these models, we prove that the enrgy has a polynomial energy
decay rate that depends on the order of the frcational derivative.
