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In this paper, we study the indirect stabilization of a system of plate equations which are weakly coupled and locally damped. By virtue of the general results due to Burq in the study of asymptotic behavior of solutions, we prove that the semigroup associated to the system is logarithmically stable under some assumptions on the damping and the coupling terms. For this purpose, we adopt an approach based on the growth of the resolvent on the imaginary axis, which can be obtained by some Carleman estimates.

International audience; In this paper, we prove a stability estimate of the conductivity parameters identification problem in cardiac electrophysiology. The propagation of the electrical wave in the heart is described by the monodomain model coupled to an elliptic equation describing the diffusion of the electrical wave in the whole body. Our result concerns both heart and torso conductivity parameters. The main difficulty that we solve in this paper is related to the transmission conditions between the heart and the torso. We first, establish Carleman estimates for the coupled heart-torso system. Then, using these estimates and the Bukhgeim and Klibanov approach, we prove a Lipschitz stability estimate of cardiac and torso conductivity parameters.

International audience; We prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data.

For a symmetric hyperbolic system of the first order, we prove a Carleman estimate under some positivity condition concerning the coefficient matrices. Next, applying the Carleman estimate, we prove an observability \begin{document}$ L^2 $\end{document}-estimate for initial values by boundary data.

On the basis of the Carleman estimate for the parabolic equation, we prove a Carleman estimate for the integro-differential operator $$\partial _t-\triangle +\int _0^t K(x,t,r)\triangle \ dr$$ where the integral kernel has a behaviour like a weakly singular one. In the proof we consider the integral term as a perturbation. The crucial point is a special choice of the time factor of the weight function.

We consider the reaction-diffusion equation with discontinuities in the diffusion coefficient and the potential term. We start by deriving the Carleman estimate for the discontinuous reaction-diffusion operator which is deployed in the inverse problems of finding the stability result of the two discontinuous coefficients from the internal observations of the given parabolic equation.

International audience; In this paper we investigate the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain, from partial measurement of the solution on the boundary. Namely, if the potential is known in a neighborhood of the boundary of the spatial domain, we prove that it can be logarithmic stably determined in the whole waveguide from a single observation of the solution on any arbitrary strip-shaped subset of the boundary.

In this article, we first establish a global Carleman estimate for an ultrahyperbolic Schrödinger equation. Next, we prove Hölder stability for the inverse problem of determining a coefficient or a source term in the equation by some lateral boundary data. © 2019, Springer Nature Switzerland AG.

We study the problem of determining uniquely a time-dependent singular potential \begin{document}$q$\end{document} , appearing in the wave equation \begin{document}$\partial_t^2u-Δ_x u+q(t,x)u = 0$\end{document} in \begin{document}$Q = (0,T)×Ω$\end{document} with \begin{document}$T>0$\end{document} and \begin{document}$Ω$\end{document} a \begin{document}$ \mathcal C^2$\end{document} bounded domain of \begin{document}$\mathbb{R}^n$\end{document} , \begin{document}$n≥2$\end{document} . We start by considering the unique determination of some general singular time-dependent coefficients. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first claiming unique determination of unbounded time-dependent coefficients, which is motivated by the problem of determining general nonlinear terms appearing in nonlinear wave equations.

International audience; We consider a 4 × 4 nonlinear reaction-diffusion system posed on a smooth domain Ω of R N (N ≥ 1) with controls localized in some arbitrary nonempty open subset ω of the domain Ω. This system is a model for the evolution of concentrations in reversible chemical reactions. We prove the local exact controllability to stationary constant solutions of the underlying reaction-diffusion system for every N ≥ 1 in any time T > 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics. The proof is based on a linearization which uses return method and an adequate change of variables that creates crossed diffusion which will be used as coupling terms of second order. The controllability properties of the linearized system are deduced from Carleman estimates. A Kakutani's fixed-point argument enables to go back to the nonlinear parabolic system. Then, we prove a global controllability result in large time for 1 ≤ N ≤ 2 thanks to our local controllabillity result together with a known theorem on the asymptotics of the free nonlinear reaction diffusion system.

International audience; We consider anisotropic elliptic and parabolic operators in a bounded stratified media in $\R^n$ characterized by discontinuties of the coefficients in one direction. The surfaces of discontinuities cross the boundary of the domain. We prove Carleman estimates for these operators with an arbitrary observation region.

We consider an inverse problem of determining a viscosity coefficient in the Navier–Stokes equation by observation data in a neighborhood of the boundary. We prove the Lipschitz stability by the Carleman estimates in Sobolev spaces of negative order.

We study the exponential stability of the plate equations with potential of second order and indefinite sign damping term. By means of a global Carleman-type estimate and the usual perturbation method, we show that the energy of the system decays exponentially provided that the negative damping is sufficiently small. Both the energy decay rate and the upper bound estimate on the negative damping are given explicitly.

The exact controllability and observability for a heat equation with hyperbolic memory kernel in anisotropic and nonhomogeneous media are considered. Due to the appearance of such a kind of memory, the speed of propagation for solutions to the heat equation is finite and the corresponding controllability property has a certain nature similar to hyperbolic equations, and is significantly different from that of the usual parabolic equations. By means of Carleman estimate, we establish a positive controllability and observability result under some geometric condition. On the other hand, by a careful construction of highly concentrated approximate solutions to hyperbolic equations with memory, we present a negative controllability and observability result when the geometric condition is not satisfied.

In a bounded domain of R n+1, n ≧ 2, we consider a second-order elliptic operator, $${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}$$ , where the (scalar) coefficient c(x) is piecewise smooth yet discontinuous across a smooth interface S. We prove a local Carleman estimate for A in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient c at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions and the Calderon projector technique. Following the method of Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) we then prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator $${{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}}$$ .

We prove a conditional stability estimate of log-type for determining unknown boundaries from a single Cauchy data taken on an accessible subboundary. Our approach relies on new interior and boundary estimates derived from the Carleman estimate for elliptic equations. A local stability result for target identification of an acoustic sound-soft scatterer from a single far-field pattern is also obtained.

International audience; We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise $\Con^1$). 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.

A parabolic equation/inequality in an infinite domain is considered. The lateral Cauchy data are given at an arbitrary C 2 -smooth lateral surface. The inverse problem of the interest of this paper consists in an estimate of the unknown initial condition via these Cauchy data.

We derive global Carleman estimates for one-dimensional linear parabolic operators ∂ t ± ∂ x ( c ∂ x ) with a coefficient c with bounded variations. These estimates are obtained by approximating c by piecewise regular coefficients, c e , and passing to the limit in the Carleman estimates associated to the operators defined with c e . Such estimates yield results of controllability to the trajectories for a class of semilinear parabolic equations. To cite this article: J. Le Rousseau, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

International audience; In this article, we prove uniqueness results for coefficient inverse problems regarding wave, heat or Schr\"odinger equation on a tree-shaped network, as well as the corresponding stability result of the inverse problem for the wave equation. The objective is the determination of the potential on each edge of the network from the additional measurement of the solution at all but one external end points. Several results have already been obtained in this precise setting or in similar cases, and our main goal is to propose a unified and simpler method of proof of some of these results. The idea which we will develop for proving the uniqueness is to use a more traditional approach in coefficient inverse problems by Carleman estimates. Afterwards, using an observability estimate on the whole network, we apply a compactness-uniqueness argument and prove the stability for the wave inverse problem.

We consider the inverse problem of determining the time independent scalar potential of the dynamic Schr\"odinger equation in an infinite cylindrical domain from one boundary Neumann observation of the solution. We prove H\"older stability by choosing the Dirichlet boundary condition suitably.

We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.

International audience; Let B be a n × n block diagonal matrix in which the first block C τ is an hermitian matrix of order (n − 1) and the second block c is a positive function. Both are piecewise smooth in Ω, a bounded domain of R n. If S denotes the set where discontinuities of C τ and c can occur, we suppose that Ω is stratified in a neighborhood of S in the sense that locally it takes the form Ω × (−δ, δ) with Ω ⊂ R n−1 , δ > 0 and S = Ω × {0}. We prove a Carleman estimate for the elliptic operator A = −∇ · (B∇) with an arbitrary observation region. This Carleman estimate is obtained through the introduction of a suitable mesh of the neighborhood of S and an associated approximation of c involving the Carleman large parameters.

This article shows that knowledge of the Dirichlet-Neumann map on certain subsets of the boundary for input functions supported roughly on the rest of the boundary can be used to determine a magnetic Schr\"{o}dinger operator. With some geometric conditions on the domain, either the subset on which the DN map is measured or the subset on which the input functions have support may be made arbitrarily small. This is an improvement on the partial data result in a paper by Dos Santos Ferreira, Kenig, Sj\"{o}strand, and Uhlmann. The method involves modifying the Carleman estimate in that paper by conjugation with operators built from pseudodifferential pieces., Comment: 47 pages, 2 figures

International audience; This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C 1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV 9 (2003) 621-635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces a nearly optimal convergence rate for the method of quasi-reversibility introduced in [Lattès and Lions, Dunod (1967)] to solve the ill-posed Cauchy problems. © EDP Sciences, SMAI, 2010.

International audience; The paper investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg-Landau equation. Zero-controllability results are derived from a new Carleman estimate and an analysis based upon the theory of sectorial operators.

The paper is devoted to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators ( α + i β ) ∂ t + ∑ j , k = 1 n ∂ k ( a j k ∂ j ) (with real functions α and β), by which we develop a universal approach, based on global Carleman estimate, to deduce not only the desired explicit observability estimate for the linearized complex Ginzburg–Landau equation, but also all the known controllability/observability results for the parabolic, hyperbolic, Schrodinger and plate equations that are derived via Carleman estimates.

By means of global Carleman-type estimate, we study the stabilization problem of the wave equations with potential and indefinite damping. The energy decay rate of the system is given explicitly. Also, we obtain an upper bound estimate on the negative damping to guarantee the energy of the system decays exponentially.