Ouahiba Zair

University of Science and Technology Houari Boumediene | usthb · Department of Analysis

In this paper, we prove controllability results for a two-dimensional semilinear heat equation with mixed boundary conditions. It is well-known that mixed boundary conditions can present a singular behaviour of the solution. First, we will prove global Carleman estimates then we will use these inequalities to obtain controllability results.

We study the boundary stabilization of the wave equation by a nonlinear feedback active on a part of the boundary in geometric situations for which the solutions have singularities. These singularities appear at the interfaces at which the mixed Neumann Dirichlet boundary conditions meet. Under a simple geometrical condition concerning the orientat...

In this work we give a null-controllability result for the semi-linear heat equation in a polygonal or cracked bounded domain of . We suppose that the nonlinearity grows slower than as and then we prove our result by using Schauder's fixed point theorem.

We consider an inverse problem of determining point sources in vibrating plate by boundary measurements. We show that the boundary observation of the domain determines uniquely the sources for an arbitrarily small time of observation and we establish a conditional stability.

It is well known that in a regular domain, the solutions of the Laplace equation with mixed boundary conditions can present a singular part. In this work, we prove a Carleman estimate for the two dimensional domain heat equation in presence of these singularities.

We consider the Cauchy problem associated to the heat equation firstly in a plane domain with a reentrant corner, then in a cracked domain. By constructing a weight function, we prove a Carleman inequality and we deduce a result of controllability.

We consider the Cauchy problem associated to the heat equation firstly in a plane domain with a reentrant corner, then in a cracked domain. By constructing a weight function, we show a result of null controllability using Carleman estimates.

We consider two inverse problems concerning the first and second Petrovsky system of determining point sources ξ k , weights d k and the number K in vibrating beams, governed by the equation ∂ t 2 u(t,x)+u (4) (x,t)=λ(t)∑ k=1 K α k δ(x-ξ k )in]0,1[×]0,T[, by boundary measurements. We show that the boundary observation at one extremity of the domain...

We consider the inverse problem of determining point wave sources in heteregeneous trees, extensions of one-dimensional stratified sets. We show that the Neumann boundary observation on a part of the lateral boundary determines uniquely the point sources if the time of observation is large enough. We further establish a conditional stability and gi...

We consider the inverse problem of determining one emerging crack in a plane heterogeneous medium. We show that one measurement on a part of the boundary with appropriate flux uniquely determines such a crack. We further establish a local Lipschitz continuity result. At the end, we give an approximation result for the nonstationary heat equation. C...