Mourad Bellassoued

Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients

One of the basic inverse problems in an anisotropic media is the determination of coefficients in a bounded domain with a single measurement. We consider the problem of finding the coefficient of the second derivatives in a second-order hyperbolic equation with variable coefficients. Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement. Moreover we show that our uniqueness results yield the Lipschitz stability estimate in L 2 space for solution to the inverse problem under consideration.One of the basic inverse problems in an anisotropic media is the determination of coefficients in a bounded domain with a single measurement. We consider the problem of finding the coefficient of the second derivatives in a second-order hyperbolic equation with variable coefficients. Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement. Moreover we show that our uniqueness results yield the Lipschitz stability estimate in L 2 space for solution to the inverse problem under consideration.

Global logarithmic stability in inverse hyperbolic problem by arbitrary boundary observation

We study the global stability in determination of a coefficient of the zeroth-order term in a second-order hyperbolic equation from data of the solution in a subboundary over a time interval. Providing regular initial data, without any assumption on the dynamics (i.e. without the geometric optics condition for the observability), we prove the uniqueness in multidimensional hyperbolic inverse problems with a single measurement. Moreover, we show that our uniqueness results yield the logarithm stability estimate in L2 space for solution of the inverse problem under consideration.

Lipschitz stability for a hyperbolic inverse problem by finite local boundary data

In this article we consider the inverse problem of determining the potential q in a wave equation in a bounded smooth domain Ω in from a finite number of data of the hyperbolic Dirichlet to Neumann map and we prove the Lipschitz stability in determining q. Our main result is stated as follows. Let T> diam Ω and Γ0⊂∂Ω. For any k-dimensional space X in L ∞(Ω), there exist 2k-functions f 1, …, f 2k on (0,T)×∂Ω such that , provided that q1, q2∈X are uniformly bounded in a suitable Sobolev space. Here Λq is the Dirichlet to Neumann map for the coefficient q(x).

Carleman estimates and an inverse heat source problem for the thermoelasticity system

According to the linear theory of thermoelasticity, we consider a bounded and isotropic body Ω whose mechanical behaviour is described by the Lame system coupled with the heat equation. Assuming null surface displacement on the whole boundary, we discuss the inverse problem of determining the heat source only by the observation of surface traction in a subdomain ω satisfying ∂ω ⊃ ∂Ω along a sufficiently large time interval. Our main result is a Holder stability estimate for the inverse problem. For it, we prove a Carleman estimate for the thermoelasticity system.

Determination of a coefficient in the wave equation with a single measurement

We consider an inverse problem of finding the coefficient of the second-order derivatives in a second-order hyperbolic equation with variable coefficients. Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement of Neumann data on a suitable sub-boundary. Moreover we show that our uniqueness yields the Lipschitz stability estimate in L 2 space for solution to the inverse problem. The key is a Carleman estimate for a hyperbolic operator with variable coefficients.

Carleman estimate with second large parameter for second order hyperbolic operators in a Riemannian manifold and applications in thermoelasticity cases

In this article we prove a Carleman estimate with second large parameter for a second order hyperbolic operator in a Riemannian manifold ℳ. Our Carleman estimate holds in the whole cylindrical domain ℳ × (0, T) independent of the level set generated by a weight function if functions under consideration vanish on boundary ∂(ℳ × (0, T)). The proof is direct by using calculus of tensor fields in a Riemannian manifold. Then, thanks to the dependency of the second larger parameter, we prove Carleman estimates also for (i) a coupled parabolic-hyperbolic system (ii) a thermoelastic plate system (iii) a thermoelasticity system with residual stress.

Inverse boundary value problem for the dynamical heterogeneous Maxwell's system

We consider the inverse problem of determining the isotropic inhomogeneous electromagnetic coefficients of the non-stationary Maxwell equations in a bounded domain of , from a finite number of boundary measurements. Our main result is a Holder stability estimate for the inverse problem, where the measurements are exerted only in some boundary components. For it, we prove a global Carleman estimate for the heterogeneous Maxwell system under boundary conditions.

Stability of the determination of the surface impedance of an obstacle from the scattering amplitude

We prove a stability estimate of logarithmic type for the inverse problem consisting in the determination of the surface impedance of an obstacle from the scattering amplitude. We present a simple and direct proof, which is essentially based on an elliptic Carleman inequality. Copyright

Stability estimate for an inverse problem for the wave equation in a magnetic field

In this article, we prove stability estimate of the inverse problem of determining the magnetic field entering the magnetic wave equation in a bounded smooth domain in ℝ d from boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet-to-Neumann map associated to the solutions to the magnetic wave equation. We prove in dimension d ≥ 2 that the knowledge of the Dirichlet-to-Neumann map for the magnetic wave equation measured on the boundary determines uniquely the magnetic field and we prove a Hölder-type stability in determining the magnetic field induced by the magnetic potential.

Lipschitz stability in an inverse problem for a hyperbolic equation with a finite set of boundary data

We consider an inverse problem of determining multiple coefficients of principal part of a scalar hyperbolic equation with Dirichlet boundary data. We prove the uniqueness and a Lipschitz stability estimate in the inverse problem with some observations on a suitable sub-boundary satisfying an appropriate geometrical condition. The key is a Carleman estimate for a hyperbolic operator with variable coefficients.