In this paper we consider the transmission problem of a material composed of three components; one of them is a Kelvin--Voigt viscoelastic material, the second is an elastic material (no dissipation), and the third is an elastic material inserted with a frictional damping mechanism. The main result of this paper is that the rate of decay will depend on the position of each component. When the viscoelastic component is not in the middle of the material, then there exists exponential stability of the solution. Instead, when the viscoelastic part is in the middle of the material, there is not exponential stability. In this case we show that the corresponding solution decays polynomially as $1/t^{2}$. Moreover we show that the rate of decay is optimal over the domain of the infinitesimal generator. Finally, using a second order scheme that ensures the decay of energy (Newmark-$\beta$ method), we give some numerical examples which demonstrate this asymptotic behavior.

Here we study an inverse problem for a quasilinear hyperbolic equation. We start by proving the existence of solutions to the problem which is posed as the minimization of a suitable cost function. Then we use a Lagrangian formulation in order to formally compute the gradient of the cost function introducing an adjoint equation. Despite the fact that the Lagrangian formulation is formal and that the cost function is not necessarily differentiable, a viscous perturbation and a numerical approximation of the problem allow us to justify this computation. When the adjoint problem for the quasi-linear equation admits a smooth solution, then the perturbed adjoint states can be proved to converge to that very solution. The sequences of gradients for both perturbed problems are also proved to converge to the same element of the subdifferential of the cost function. We evidence these results for a large class of numerical schemes and particular cost functions which can be applied to the identification of isotherms for conservation laws modeling distillation or chromatography. They are illustrated by numerical examples.