Uniform exponential decay for viscous damped systems

We consider a class of viscous damped vibrating systems. We prove that, under the assumption that the damping term ensures the exponential decay for the corresponding inviscid system, then the exponential decay rate is uniform for the viscous one, regardless what the value of the viscosity parameter is. Our method is mainly based on a decoupling argument of low and high frequencies. Low frequencies can be dealt with because of the effectiveness of the damping term in the inviscid case while the dissipativity of the viscous term guarantees the decay of the high frequency components. This method is inspired in previous work by the authors on time-discretization schemes for damped systems in which a numerical viscosity term needs to be added to ensure the uniform exponential decay with respect to the time-step parameter.