Asymptotic Analysis - Volume 71, issue 1-2

Authors: Ervedoza, Sylvain

Article Type: Research Article

Abstract: In this article, we derive uniform admissibility and observability properties for the finite element space semi-discretizations of iż=A0 z, where A0 is an unbounded self-adjoint positive definite operator with compact resolvent. In order to address this problem, we present several spectral criteria for admissibility and observability of such systems, which will be used to derive several results for space semi-discretizations of iż=A0 z. Our approach provides very general results, which stand in any dimension and for any regular mesh (in the sense of finite elements). We also present applications to admissibility and observability for fully discrete approximation schemes, and …to controllability and stabilization issues. Show more

Keywords: observability, admissibility, Schrödinger equation, finite element method, resolvent estimates, controllability, stabilization

DOI: 10.3233/ASY-2010-1028

Citation: Asymptotic Analysis, vol. 71, no. 1-2, pp. 1-32, 2011

Authors: Ould-Hammouda, Amar

Article Type: Research Article

Abstract: We consider elliptic problems in periodically perforated domains in RN , N≥3, with nonhomogeneous Neumann conditions on the boundary of the holes. The aim is to give the asymptotic behavior of the solutions as the period ε goes to zero. Two geometries are considered. In the first one, all the holes are “small”, i.e., their size is of order of εr(ε) with r(ε)→0. The second geometry is more general, there are small holes as before but also holes of size of the order of ε (the last ones corresponding to the classical homogenization situation). Our study is performed by the …periodic unfolding method from C. R. Acad. Sci. Paris Ser. I 335 (2002), 99–104, adapted to the case of holes of size εr(ε) (see J. Math. Pures Appl. 89 (2008), 248–277). The use of this method allows us to study second-order operators with highly oscillating coefficients and so, to generalize here the results of RAIRO Modél. Math. Anal. Numér. 4(22) (1988), 561–608. In both cases, if r(ε)=exp (N/N−1), an additional term appears in the right-hand side of the limit equation. Show more

Keywords: homogenization, periodic unfolding method, perforated domains, small holes

DOI: 10.3233/ASY-2010-1010

Citation: Asymptotic Analysis, vol. 71, no. 1-2, pp. 33-57, 2011

Authors: Lablée, Olivier

Article Type: Research Article

Abstract: The aim of this paper is to study the semi-classical behaviour of Schrödinger's dynamics for an one-dimensional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. This classical motion is periodic and the period Thyp is order of |ln h|. And, for large time, a new period Trev for the quantum dynamics appears: the initial wave packets form again at t=Trev . Moreover, for the time t=p/qTrev a fractional revivals phenomenon of the …initial wave packets appears: there is a formation of a finite number of clones of the original wave packet. Show more

Keywords: Schrödinger's dynamics, revivals of wave packets, semi-classical analysis, hyperbolic trajectory, Schrödinger operator with double wells potential

DOI: 10.3233/ASY-2010-1012

Citation: Asymptotic Analysis, vol. 71, no. 1-2, pp. 59-99, 2011

Authors: Ben Belgacem, Faker | Bernardi, Christine | El Fekih, Henda

Article Type: Research Article

Abstract: We are interested in the optimal control problem of a parabolic equation with no state constraints, where the quadratic cost functional involves a final observation and the control variable is a Dirichlet boundary condition. Practical considerations lead us to use Dirichlet controls with no more regularity than square integrability, which arise some technical difficulties in the mathematical analysis. After setting the state equation and the related adjoint equation we fit the two coupled equations into a mixed space–time variational framework. The resulting saddle-point problem turns out to be well posed. We then use a Robin penalization on the Dirichlet control …which enables us to re-transcript the mixed problem in a form better suited to numerical computations. We analyze and establish the convergence when the penalty parameter tends to zero, first without additional smoothness assumptions on the optimal control and then for smooth controls. Show more

Keywords: Dirichlet control, mixed variational formulation, penalized Robin control, convergence analysis

DOI: 10.3233/ASY-2010-1015

Citation: Asymptotic Analysis, vol. 71, no. 1-2, pp. 101-121, 2011