Asymptotic Analysis

We are considering Ingham's inequality (see [8]) for a family of exponential functions {e^iλ_nt}_n≥1, in the case in which the distance |λ_n+1−λ_n| between two consecutive exponents becomes smaller and smaller for |n|≤N but there still exists an asymptotic gap sufficiently large. We give explicit estimates for the two constants appearing in the inequality and we analyze how does the small gap between the first exponents affect the constants. These results are applied to a control problem for the wave equation in a case in which the geometric condition for controllability, deduced in [4], are not satisfied.

We study the large time behavior of the solutions of a homogenous string equation with a homogenous Dirichlet boundary condition at the left end and a homogenous Neuman boundary condition at the right end. A pointwise interior actuator gives a linear viscous damping term. We give a complete characterization of the positions of the actuator for which the system becomes exponentially stable in the energy space. In the case of nonexponential decay in the energy space we give explicit polynomial decay estimates valid for regular initial data. Moreover we show that the fastest decay rate is obtained if the actuator is located at the middle point of the string.

A mixed Dirichlet–Ventcel problem with a small parameter in a polygonal domain of the plane is considered. Since the limit problem is a mixed Dirichlet–Neumann problem, we are in presence of a singular perturbed problem. We give the full asymptotic of the solution of the Dirichlet–Ventcel problem with the help of outer terms and boundary layer terms. Here the main difficulty relies on the singular behaviour of the solutions near the corner points where the boundary conditions change, so that the boundary layer terms are corner layer terms.

We show that the energy of a discrete system of material points on a line and subjected to random nearest‐neighbour interactions, almost surely converges in a variational sense to a deterministic energy defined on spaces of functions with bounded variation.

Résumé. Nous montrons que l'énergie discrète en dimension 1 d'un système de points matériels soumis à des intéractions aléatoires locales, converge presque surement, en un sens variationnel, vers une énergie déterministe définie dans les espaces BV ou SBV.

We investigate the behaviour of the solutions of kinetic equations as a parameter related to both the mean free path and a characteristic length of heterogeneities goes to 0. We obtain in the limit a diffusion equation for the macroscopic density which combines the homogenization effects to the diffusion approximation. In particular the limit equation contains drift terms related to the behaviour of the kernel of the collision operator.