Fatiha Alabau

Boundary Observability, Controllability, and Stabilization of Linear Elastodynamic Systems

In 1988 Lions obtained observability and exact controllability results for linear homogeneous isotropic elastodynamic systems [ SIAM Rev., 30 (1988), pp. 1--68]. Applying some new identities we extend his theorems to nonisotropic systems. In 1991 Lagnese obtained uniform stabilizability results for two-dimensional linear homogeneous isotropic systems by applying a somewhat artificial feedback [Nonlinear Anal., 16 (1991), pp. 35--54]. Then he asked whether analogous results hold for a natural and physically implementable boundary feedback. Using some new identities and applying a method introduced in 1987 by Zuazua and the second author [ J. Math. Pures. Appl., 69 (1990), pp. 33--54], we give an affirmative answer to this question in all dimensions and also for nonisotropic systems. Moreover, we obtain good decay estimates. Finally, applying a recent general method of uniform stabilization, we construct boundary feedbacks leading to arbitrarily large energy decay rates.

Structural properties of the one-dimensional drift-diffusion models for semiconductors

This paper is devoted to the analysis of the one-dimensional current and voltage drift-diffusion models for arbitrary types of semiconductor devices and under the assumption of vanishing generation recombination. We show in the course of this paper, that these models satisfy structural properties, which are due to the particular form of the coupling of the involved systems. These structural properties allow us to prove an existence and uniqueness result for the solutions of the current driven model together with monotonicity properties with respect to the total current I, of the electron and hole current densities and of the electric field at the contacts. We also prove analytic dependence of the solutions on I. These results allow us to establish several qualitative properties of the current voltage characteristic. In particular, we give the nature of the (possible) bifurcation points of this curve, we show that the voltage function is an analytic function of the total current and we characterize the asymptotic behavior of the currents for large voltages. As a consequence, we show that the currents never saturate as the voltage goes to +00, contrary to what was predicted by numerical simulations by M. S. Mock (Compel. 1 (1982), pp. 165-174). We also analyze the drift-diffusion models under the assumption of quasi-neutral approximation. We show, in particular, that the reduced current driven model has at most one solution, but that it does not always have a solution. Then, we compare the full and the reduced voltage driven models and we show that, in general, the quasi-neutral approximation is not accurate for large voltages, even if no saturation phenomenon occurs. Finally, we prove a local existence and uniqueness result for the current driven model in the case of small generation recombination terms.

A uniqueness theorem for reverse biased diodes

We analyze the basic semiconductor device equations in the case of a symmetric one-dimensional reverse biased diode. The “classical” proofs of uniqueness theorems for this nonlinear system are based on asynlptotic methods and are not valid for arbitrary values of the applied bias. The main result of this paper is that every symmetric solution of this system is locally unique for any reverse bias value. Our method is based oq a “decoupling” of the associated linearized system and on the generalized maximum principle.

On the existence of multiple steady-state solutions in the theory of electrodiffusion. Part I: the nonelectroneutral case. Part II: a constructive method for the electroneutral case

We give a constructive method for giving examples of doping functions and geometry of the device for which the nonelectroneutral voltage driven equations have multiple solutions. We show in particular, by performing a singular perturbation analysis of the current driven equations that if the electroneutral voltage driven equations have multiple solutions then the nonelectroneutral voltage driven equations have multiple solutions for sufficiently small normed Debye length. We then give a constructive method for giving examples of data for which the electroneutral voltage driven equations have multiple solutions.