Assia Benabdallah - Academia.edu (original) (raw)

Papers by Assia Benabdallah

We consider the heat equation with a discontinuous diusion coecien t and give uniqueness and stab... more We consider the heat equation with a discontinuous diusion coecien t and give uniqueness and stability results for both the diusion coecien t and the initial condition from a measurement of the solution on an ar- bitrary part of the boundary and at some arbitrary positive time. The key ingredient is the derivation of a Carleman-type estimate. The diu- sion

Aone-dimensional VonKarmanmodelwiththermal eectsisstudied. We derive the equations that constitut... more Aone-dimensional VonKarmanmodelwiththermal eectsisstudied. We derive the equations that constitute the mathematical model, and proveexistenceanduniquenessofaglobalsolution. ThenusingLyapunov functions, we show that solutions decay exponentially.

Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis

We look for dynamical stabilizers for a given second order equa- tion. Sucient conditions on the ... more We look for dynamical stabilizers for a given second order equa- tion. Sucient conditions on the stabilizers are obtained. A complete descrip- tion of a one-parameter family of such stabilizers is given which includes a precise behaviour of the spectrum of the associated operators. These abstract results are applied to thermoelastic systems.

Mathematical biosciences, 2009

In cancer diseases, the appearance of metastases is a very pejorative forecast. Chemotherapies ar... more In cancer diseases, the appearance of metastases is a very pejorative forecast. Chemotherapies are systemic treatments which aim at the elimination of the micrometastases produced by a primitive tumour. The efficiency of chemotherapies closely depends on the protocols of administration. Mathematical modeling is an invaluable tool to help in evaluating the best treatment strategy. Iwata et al. [K. Iwata, K. Kawasaki, N. Shigesad, A dynamical model for the growth and size distribution of multiple metastatic tumors, J. Theor. Biol. 203 (2000) 177.] proposed a partial differential equation (PDE) that describes the metastatic evolution of an untreated tumour. In this article, we conducted a thorough mathematical analysis of this model. Particularly, we provide an explicit formula for the growth rate parameter, as well as a numerical resolution of this PDE. By increasing our understanding of the existing model, this work is crucial for further extension and refinement of the model. It set...

Cancer research, Jan 15, 2014

Defining tumor stage at diagnosis is a pivotal point for clinical decisions about patient treatme... more Defining tumor stage at diagnosis is a pivotal point for clinical decisions about patient treatment strategies. In this respect, early detection of occult metastasis invisible to current imaging methods would have a major impact on best care and long-term survival. Mathematical models that describe metastatic spreading might estimate the risk of metastasis when no clinical evidence is available. In this study, we adapted a top-down model to make such estimates. The model was constituted by a transport equation describing metastatic growth and endowed with a boundary condition for metastatic emission. Model predictions were compared with experimental results from orthotopic breast tumor xenograft experiments conducted in Nod/Scidγ mice. Primary tumor growth, metastatic spread and growth were monitored by 3D bioluminescence tomography. A tailored computational approach allowed the use of Monolix software for mixed-effects modeling with a partial differential equation model. Primary tu...

ESAIM: Proceedings, 2000

ABSTRACT The full von Karman system accounting for in plane acceleration and nonlinear thermal ef... more ABSTRACT The full von Karman system accounting for in plane acceleration and nonlinear thermal effects is considered. The results obtained in this paper: (i) wellposedness of regular and weak (finite energy) solutions, (ii) uniform decay rates of energy function, extend those obtained earlier in (A. Benabdallah and I.Lasiecka) assia-las for a more restrictive model which does not account for the nonlinear thermal coupling.

ESAIM: Proceedings, 1997

ABSTRACT The aim of this paper is the study of dynamical stabilization for distributed systems. F... more ABSTRACT The aim of this paper is the study of dynamical stabilization for distributed systems. For a given system, we describe classes of strong and uniform dynamical stabilizers. Applications to physical models are given.

SIAM Journal on Control and Optimization, 2003

In this article, we study the controllability to the trajectories of 2 × 2 nonlinear parabolic sy... more In this article, we study the controllability to the trajectories of 2 × 2 nonlinear parabolic systems for control forces acting on a single equation of the system. This result, which in particular applies to Caginalp's phase-field model, actually extends those obtained for the semilinear heat equations. The proof relies on Kakutani's fixed point theorem and makes use of an observability estimate for the associated linearized system.

SIAM Journal on Control and Optimization, 2007

CVD diamond is a radiation hard sensor material which may be used for charged particle tracking n... more CVD diamond is a radiation hard sensor material which may be used for charged particle tracking near the interaction region in experiments at high luminosity colliders. The goal of the work described here is to investigate the use of several detector planes made of CVD diamond strip sensors for charged particle tracking. Towards this end a tracking telescope composed entirely of CVD diamond planes has been constructed. The telescope was tested in muon beams and its tracking capability has been investigated.

Journal of Mathematical Analysis and Applications, 2006

This work is concerned with the null-controllability of semilinear parabolic systems by a single ... more This work is concerned with the null-controllability of semilinear parabolic systems by a single control force acting on a subdomain.

Journal of Mathematical Analysis and Applications, 2007

We study the observability and some of its consequences (controllability, identification of diffu... more We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise C 1 ). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.

Journal of Functional Analysis, 2011

... Assia Benabdallah a , Yves Dermenjian a and Jérôme Le Rousseau b , low asterisk , E-mail The ... more ... Assia Benabdallah a , Yves Dermenjian a and Jérôme Le Rousseau b , low asterisk , E-mail The Corresponding Author. ... term (ie, the action of a the first-order operator u |S ) yields a negative contribution, unless a monotonicity assumption on the coefficient c is ade as in [12]. ...

Journal of Evolution Equations, 2009

We present a generalization of the Kalman rank condition to the case of n × n linear parabolic sy... more We present a generalization of the Kalman rank condition to the case of n × n linear parabolic systems with constant coefficients and diagonalizable diffusion matrix. To reach the result, we are led to prove a global Carleman estimate for the solutions of a scalar 2n-order parabolic equation and deduce from it an observability inequality for our adjoint system.

Journal of Differential Equations, 2000

The full von Karman system accounting for in plane acceleration and thermal effects is considered... more The full von Karman system accounting for in plane acceleration and thermal effects is considered. The main results of the paper are: (i) the wellposedness of regular and weak (finite energy) solutions, (ii) the uniform decay rates obtained for the energy function in the presence of mechanical damping affecting only the solenoidal part of the velocity field representing the horizontal displacements of the plate. The obtained decay rates are uniform with respect to the parameter # which represents the momenta of inertia and whose presence distinguishes the``paraboliclike'' from the hyperbolic character of the dynamics.

Journal of Differential Equations, 2003

Linear systems of Timoshenko type equations for beams including a memory term are studied. The ex... more Linear systems of Timoshenko type equations for beams including a memory term are studied. The exponential decay is proved for exponential kernels, while polynomial kernels are shown to lead to a polynomial decay. The optimality of the results is also investigated. r

ESAIM: Control, Optimisation and Calculus of Variations, 2005

ABSTRACT We study the null controllability by one control force of some linear systems of parabol... more ABSTRACT We study the null controllability by one control force of some linear systems of parabolic type. We give sufficient conditions for the null controllability property to be true and, in an abstract setting, we prove that it is not always possible to control.

Differential Equations & Applications, 2009

In this paper we present a generalization of the Kalman rank condition for linear ordinary differ... more In this paper we present a generalization of the Kalman rank condition for linear ordinary differential systems to the case of systems of n coupled parabolic equations (posed in the time interval (0,T ) with T > 0 ) where the coupling matrices A and B depend on the time variable t . To be precise, we will prove that the Kalman rank condition rank [A|B](t 0 ) = n , with t 0 ∈ [0,T ] , is a sufficient condition (but not necessary) for obtaining the exact controllability to the trajectories of the considered parabolic system. In the case of analytic matrices A and B (and, in particular, constant matrices), we will see that the Kalman rank condition characterizes the controllability properties of the system. When the matrices A and B are constant and condition rank [A |B] = n holds, we will be able to state a Carleman inequality for the corresponding adjoint problem.

Comptes Rendus Mécanique, 2006

We study the observability and some of its consequences for the one-dimensional heat equation wit... more We study the observability and some of its consequences for the one-dimensional heat equation with a discontinuous coefficient (piecewise C 1 ). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields results of controllability to the trajectories for semilinear equations. It also yields a stability result for the inverse problem of the identification of the diffusion coefficient. To cite this article: A. Benabdallah, Y. Dermenjian, J. Le Rousseau, C. R. Mécanique -(2006).

Comptes Rendus Mathematique, 2007

In this article, we study the controllability of a class of parabolic systems of the form Yt = (D... more In this article, we study the controllability of a class of parabolic systems of the form Yt = (D∆+A)Y +Bχωu with Dirichlet conditions on the boundary of a bounded domain Ω, where ω ⊂ Ω is a subdomain. Here D, A ∈ L(R n ), B ∈ L(R m ; R n ) and we prove that the algebraic Kalman condition extends to such systems. To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser.