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The main concern of this article is to investigate the boundary controllability of some $$2\times 2$$ one-dimensional parabolic systems with both the interior and boundary couplings: The interior coupling is chosen to be linear with constant coefficient while the boundary one is considered by means of some Kirchhoff-type condition at one end of the domain. We consider here the Dirichlet boundary control acting only on one of the two state components at the other end of the domain. In particular, we show that the controllability properties change depending on which component of the system the control is being applied. Regarding this, we point out that the choices of the interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce the positive or negative controllability results. Further to this, we pursue a numerical study based on the well-known penalized HUM approach. We make some discretization for a general interior-boundary coupled parabolic system, mainly to incorporate the effects of the boundary couplings into the discrete setting. This allows us to illustrate our theoretical results as well as to experiment some more examples which fit under the general framework, for instance a similar system with a Neumann boundary control on either one of the two components.

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; In this paper, we analyze the longtime behavior of the wave equation with local Kelvin-Voigt Damping. Through introducing proper class symbol and pseudo-diff-calculus, we obtain a Carleman estimate, and then establish an estimate on the corresponding resolvent operator. As a result, we show the logarithmic decay rate for energy of the system without any geometric assumption on the subdomain on which the damping is effective.

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.

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.

In this paper, we address an inverse problem for the Korteweg-de Vries equation posed on a bounded domain with boundary conditions proposed by Colin and Ghidaglia. More precisely, we retrieve the principal coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability of this inverse problem relies on a Carleman estimate for the linearized Korteweg-de Vries equation and the Bukhgeı̌m-Klibanov method.

We study the inverse problem of determining a magnetic Schrodinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. In this context we prove the unique recovery of the magnetic field and the electric potential associated with general bounded and non-compactly supported electromagnetic potentials. By assuming that the electromagnetic potentials are known on the neighborhood of the boundary outside a compact set, we even prove the unique determination of the magnetic field and the electric potential from measurements restricted to a bounded subset of the infinite boundary. Finally, in the case of a waveguide taking the form of an infinite cylindrical domain, we prove the recovery of the magnetic field and the electric potential from partial data corresponding to restriction of Neumann boundary measurements to slightly more than half of the boundary. We establish all these results by mean of a new class of complex geometric optics solutions and of Carleman estimates suitably designed for our problem stated in an unbounded domain and with bounded electromagnetic potentials.

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.

arXiv admin note: text overlap with arXiv:1501.01384; International audience; This paper is devoted to the reconstruction of the conductivity coefficient for a nonautonomous hyperbolic operator an infinite cylindrical domain. Applying a local Carleman estimate, we prove the uniqueness and a Hölder stability in the determination of the conductivity using a single measurement data on the lateral boundary. Our numerical examples show good reconstruction of the location and contrast of the conductivity function in 3 dimensions.

Proceedings of PIERS 2017, St. Petersburg, Russia, May 22-25; International audience; This paper is devoted to the reconstruction of the time and space-dependent coefficient in an inverse hyperbolic problem in a bounded domain. Using a local Carleman estimate we prove the uniqueness and a Hölder stability in the determining of the conductivity by a single measurement on the lateral boundary. Our numerical examples show possibility of the determination of the location and the large contrast of the space-dependent function in three dimensions.

International audience; A Carleman estimate for a differential operator $P$ is a weighted energy estimate with a weight of exponential form $\exp(\tau \varphi)$ that involves a large parameter, $\tau>0$. The function $\varphi$ and the operator $P$ need to fulfill some sub-ellipticity properties that can be achieved for instance by choosing $\varphi = \exp(\alpha \psi)$, involving a second large parameter, $\alpha>0$, with $\psi$ satisfying some geometrical conditions. The purpose of this article is to give the framework to keep explicit the dependency upon the two large parameters in the resulting Carleman estimates. The analysis is absed on the introduction of a proper Weyl-Hörmander calculus for pseudo-differential operators. Carleman estimates of various strengths are considered: estimates under the (strong) \pcty condition and estimates under the simple characteristic property. In each case the associated geometrical conditions for the function $\psi$ is proven necessary and sufficient. In addition some optimality results with respect to the power of the large parameters are proven.

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.

We consider the unique recovery of a non-compactly supported and non-periodic perturbation of a Schrödinger operator in an unbounded cylindrical domain, also called waveguide, from boundary measurements. More precisely, we prove recovery of a general class of electric potentials from the partial Dirichlet-to-Neumann map, where the Dirichlet data is supported on slightly more than half of the boundary and the Neumann data is taken on the other half of the boundary. We apply this result in different contexts including recovery of some general class of non-compactly supported coefficients from measurements on a bounded subset and recovery of an electric potential, supported on an unbounded cylinder, of a Schrödinger operator in a slab.

This paper is focused on the study of an inverse problem for a non-self-adjoint hyperbolic equation. More precisely, we attempt to stably recover a first order coefficient appearing in a wave equation from the knowledge of Neumann boundary data. We show in dimension n greater than two, a stability estimate of H{\"o}lder type for the inverse problem under consideration. The proof involves the reduction to an auxiliary inverse problem for an electromagnetic wave equation and the use of an appropriate Carleman estimate.

International audience; We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove Hölder stability with the aid of a Carleman estimate specifically designed for hyperbolic waveguides.

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.

In (0,T)×Ω, Ω open subset of ℝ n , n≥2, we consider a parabolic operator P=∂ t −∇ x δ(t,x)∇ x , where the (scalar) coefficient δ(t,x) is piecewise smooth in space yet discontinuous across a smooth interface S. We prove a global in time, local in space Carleman estimate for P 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 δ at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface. In the high-frequency regime we use Calderon projectors. In the low-frequency regime we follow a more classical approach. Because of the parabolic nature of the problem we need to introduce Weyl-Hormander anisotropic metrics, symbol classes and pseudo-differential operators. Each frequency regime and the associated technique require a different calculus. A global in time and space Carleman estimate on (0,T)×M, M a manifold, is also derived from the local result.

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 $${{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}}$$ .

Motivated by several examples coming from physics, biology, and economics, we consider a class of parabolic operators that degenerate at the boundary of the space domain. We study null controllability by a locally distributed control. For this purpose, a specific Carleman estimate for the solutions of degenerate adjoint problems is proved. To cite this article: P. Cannarsa et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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.

On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called "clamped" boundary conditions , that is, homogeneous Dirichlet and Neumann conditions simultaneously. We also prove a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions. The spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as exp(C$\mu$ 1/4) where $\mu$ is the largest eigenvalue associated with the eigenfunctions appearing in the sum. This type of inequality is known for the Laplace operator. As an application, we obtain a null-controllability result for a higher-order parabolic equation. The resolvent estimate provides the spectral behavior of the plate semigroup generator on the imaginary axis. This type of estimate is known in the case of the damped wave semigroup. As an application , we deduce a stabilization result for the damped plate equation, with a log-type decay. The proofs of both the spectral inequality and the resolvent estimate are based on the derivation of different types of Carleman estimates for an elliptic operator related to the bi-Laplace operator: in the interior and at some boundaries. One of these estimates exhibits a loss of one full derivative. Its proof requires the introduction of an appropriate semi-classical calculus and a delicate microlocal argument.

We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function $u\in W^{2,2}$ with support in a punctured ball of arbitrary radius. The novelty of this Carleman estimate is that we establish an explicit dependence on the Lipschitz and ellipticity constants, the dimension of the space and the radius of the ball. In particular we provide a uniform and quantitative bound on the weight function for a class of elliptic operators given explicitly in terms of ellipticity and Lipschitz constant., Comment: 23 pages

International audience; We consider elliptic operators with complex coefficients and we derive microlocal and local Carleman estimates near a boundary, under sub-ellipticity and strong Lopatinskii conditions. Carleman estimates are weighted {\em a priori} estimates for the solutions of the associated elliptic boundary problem. The weight is of exponential form, $\exp(\tau \varphi)$, where $\tau>0$ is meant to be taken as large as desired. Such estimates have numerous applications in unique continuation, inverse problems, and control theory. Based on inequalities for interior and boundary differential quadratic forms, the proof relies on the microlocal factorization of the symbol of the conjugated operator in connection with the sign of the imaginary part of its roots. We further consider weight functions of the form $\varphi= \exp(\gamma \psi)$, with $\gamma>0$ meant to be taken as large as desired, and we derive Carleman estimates where the dependency upon the two large parameters, $\tau$ and $\lambda$, is made explicit. Applications on unique continuation properties are given.

We present new scale-free quantitative unique continuation principles for Schrödinger operators. They apply to linear combinations of eigenfunctions corresponding to eigenvalues below a predescribed energy, and can be formulated as an uncertainty principle for spectral projectors. This extends recent results of Rojas- Molina & Veselić, and Klein. We apply the scale- free unique continuation principle to obtain a Wegner estimate for a random Schrödinger operator of breather type. It holds for arbitrarily high energies. Schrödinger operators with random breather potentials have a non-linear dependence on random variables. We explain the challanges arising from this non-linear dependence. could be naturally implemented. Based on this criterion, a discretization procedure is constructed for the calculation of the optimal damping coefficient. If the internal damping is present, we show that this procedure can be used to obtain the optimal damping operator in the case of optimization over the set of all admissible damping operators.

In this paper, the authors consider inverse problems of determining a coefficient or a source term in an ultrahyperbolic equation by some lateral boundary data. The authors prove Hölder estimates which are global and local and the key tool is Carleman estimate. © 2014 Fudan University and Springer-Verlag Berlin Heidelberg., Council for Higher Education: 16.01.2012:558–2233, Manuscript received July 16, 2013. Revised November 8, 2013. 1Department of Mathematics, Bülent Ecevit University, Zonguldak 67100, Turkey. E-mail: f.golgeleyen@beun.edu.tr 2Department of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan. E-mail: myama@ms.u-tokyo.ac.jp *This work was supported by the Council of Higher Education of Turkey (No. 16.01.2012:558–2233).

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 Neumann boundary observation of the solution. Assuming that this potential is known outside some fixed compact subset of the waveguide, we prove that it may be Lipschitz stably retrieved by choosing the Dirichlet boundary condition of the system suitably. Since the proof is by means of a global Carleman estimate designed specifically for the Schr\"odinger operator acting in an unbounded cylindrical domain, the Neumann data is measured on an infinitely extended subboundary of the cylinder.

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.

International audience; This paper is dedicated to approximate controllability for Grushin equation on the rectangle (x, y) ∈ (−1, 1) × (0, 1) with an inverse square potential. This model corresponds to the heat equation for the Laplace-Beltrami operator associated to the Grushin metric on R^2 , studied by Boscain and Laurent. The operator is both degenerate and singular on the line {x = 0}. The approximate controllability is studied through unique continuation of the adjoint system. For the range of singularity under study, approximate controllability is proved to hold whatever the degeneracy is. Due to the internal inverse square singularity, a key point in this work is the study of well-posedness. An extension of the singular operator is designed imposing suitable transmission conditions through the singular-ity. Then, unique continuation relies on the Fourier decomposition of the 2d solution in one variable and Carleman estimates for the 1d heat equa-tion solved by the Fourier components. The Carleman estimate uses a suitable Hardy inequality.