Carleman Estimates Research Articles

Insensitizing controls for the parabolic equations with dynamic boundary conditions

This paper concerns with the insensitizing controllability property of the semilinear parabolic equation with dynamic boundary conditions. This problem can be reduced to a (relaxed) controllability problem for a coupled parabolic system with dynamic boundary conditions. An observability estimate for the corresponding coupled system with this type of boundary conditions is established, whose proof relies on a new global Carleman estimate.

Carleman estimates for the time-fractional advection-diffusion equations and applications

In this article, we prove Carleman estimates for the generalized time-fractional advection-diffusion equations by considering the fractional derivative as perturbation for the first order time-derivative. As a direct application of the Carleman estimates, we show a conditional stability estimate for a lateral Cauchy problem for the time-fractional advection-diffusion equation, and we also investigate the stability of an inverse source problem.

An inverse problem for an electroseismic model describing the coupling phenomenon of electromagnetic and seismic waves

The electroseismic model describes the coupling phenomenon of the electromagnetic waves and seismic waves in fluid immersed porous rock. Electric parameters have better contrast than elastic parameters while seismic waves provide better resolution because of the short wavelength. The combination of theses two different waves is prominent in oil exploration. Under some assumptions on the physical parameters, we derived a Hölder stability estimate to the inverse problem of recovery of the electric parameters and the coupling coefficient from the knowledge of the fields in a small open domain near the boundary. The proof is based on a Carleman estimate of the electroseismic model.

Strong unique continuation for two-dimensional anisotropic elliptic systems

In this paper, we give the strong unique continuation property for a general two dimensional anisotropic elliptic system with real coefficients in a Gevrey class under the assumption that the principal symbol of the system has simple characteristics. The strong unique continuation property is derived by obtaining some Carleman estimate. The derivation of the Carleman estimate is based on transforming the system to a larger second order elliptic system with diagonal principal part which has complex coefficients.

Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data

We propose a new numerical method to reconstruct the isotropic electrical conductivity from measured restricted Dirichlet-to-Neumann map data in electrical impedance tomography (EIT). ‘Restricted Dirichlet-to-Neumann (DtN) map data’ means that the Dirichlet and Neumann boundary data for EIT are generated by a point source running either along an interval of a straight line or along a curve located outside of the domain of interest. We ‘convexify’ the problem via constructing a globally strictly convex Tikhonov-like functional using a Carleman weight function. In particular, two new Carleman estimates are established. Global convergence to the correct solution of the gradient projection method for this functional is proven. Numerical examples demonstrate a good performance of this numerical procedure.

An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method

We propose in this paper a numerical method to solve a linear inverse source problem for general hyperbolic equations. This is the problem of reconstructing sources from the lateral Cauchy data of the wave field on the boundary of a domain. In order to achieve the goal, we derive an equation involving a Volterra integral, whose solution directly provides the desired solution of the inverse source problem. Due to the presence of such a Volterra integral, this equation is not in a standard form of partial differential equations. We employ the quasi-reversibility method to find its regularized solution. Using Carleman estimates, we show that the obtained regularized solution converges to the true solution with the Lipschitz-like convergence rate as the measurement noise tends to 0. This is one of the novelties of this paper since currently, convergence results for the quasi-reversibility method are only known for purely differential equations. Numerical tests demonstrate a good reconstruction accuracy.

Unique continuation for a reaction-diffusion system with cross diffusion

Abstract This paper concerns unique continuation for a reaction-diffusion system with cross diffusion, which is a drug war reaction-diffusion system describing a simple dynamic model of a drug epidemic in an idealized community. We first establish a Carleman estimate for this strongly coupled reaction-diffusion system. Then we apply the Carleman estimate to prove the unique continuation, which means that the Cauchy data on any lateral boundary determine the solution uniquely in the whole domain.

Null controllability for a class of degenerate parabolic equations with the gradient terms

This paper concerns a class of control systems governed by degenerate parabolic equations with the gradient terms, which are independent of the diffusion terms. The Carleman estimates and the observability inequalities for the equations are established when the degeneracy is relatively weak. Subsequently, it is proved that the control systems are null controllable. Moreover, the result can be generalized to the semilinear equations by using the fixed point theorem.

Carleman estimate for the Schrödinger equation and application to magnetic inverse problems

Abstract In this paper, we prove Lipschitz stable determination of the complex-valued electric potential and the direction of the magnetic field appearing in the dynamic Schrodinger equation with static coefficients, by finitely many partial boundary measurements of the solution. This is by means of the Bukhgeim–Klibanov method, based on an appropriate Carleman estimate. Since the time symmetrization of the static magnetic Schrodinger equation around t = 0 is not possible, we preliminarily establish a Carleman inequality specifically designed for this problem.

An inverse boundary problem for fourth-order Schrödinger equations with partial data

Abstract In this paper, we show that in dimension n ≥ 3, the knowledge of the Cauchy data for the fourth-order Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. The proof is based on the Carleman estimates and the construction of complex geometrical optics solutions.

A Carleman estimate for the linear magnetoelastic waves system and an inverse source problem in a bounded conductive medium

ABSTRACT In this paper, we consider an inverse problem for the simultaneous diffusion process of elastic and electromagnetic waves in an isotropic heterogeneous elastic body which is identified with an open bounded domain. From the mathematical point of view, the system under consideration can be viewed as the coupling between the hyperbolic system of elastic waves and a parabolic system for the magnetic field. We study an inverse problem of determining the external source terms by observations data in a neighborhood of the boundary and we prove the Hölder stability. For the proof, we show a Carleman estimate for the displacement and the magnetic field of the magnetoelastic system.

A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media

A new numerical method to solve an inverse source problem for the Helmholtz equation in inhomogenous media is proposed. This method reduces the original inverse problem to a boundary value problem for a coupled system of elliptic PDEs, in which the unknown source function is not involved. The Dirichlet boundary condition is given on the entire boundary of the domain of interest and the Neumann boundary condition is given on a part of this boundary. To solve this problem, the quasi-reversibility method is applied. Uniqueness and existence of the minimizer are proven. A new Carleman estimate is established. Next, the convergence of those minimizers to the exact solution is proven using that Carleman estimate. Results of numerical tests are presented.

Inverse Problem and Trajectory Controllability for the Swift-hohenberg Equation

This article studies an internal type inverse problem that consists of recovering a nonlinear coefficient in the Swift-Hohenberg (SH) equation and establish the controllability to the trajectories of the corresponding control system. First, we establish a Carleman estimates for the linearized problem with Neumann type boundary conditions by internal observation. Applying this Carleman estimate and regularity results, we prove stability estimate of SH equation. Further, by using the fixed point arguments, we prove the null controllability of a linearized SH equation using Carleman estimate and observability inequality derived for adjoint equation. Finally, we extend it for the trajectory controllability of the nonlinear SH equation.

Carleman Estimates for Some Two-Dimensional Degenerate Parabolic PDEs and Applications

This paper deals with a class of two-dimensional degenerate parabolic equations in a square. The goals are to obtain Carleman estimates and then establish related controllability results. We assume that the degeneracy occurs only on a part of the boundary. We present well-posedness results and then, under some geometrical assumptions, we build suitable weight functions that allow us to deduce global Carleman estimates. As an application, we prove several null controllability and Stackelberg--Nash null controllability results.

Exact Controllability of a Linear Korteweg--de Vries Equation by the Flatness Approach

We consider a linear Korteweg-de Vries equation on a bounded domain with a left Dirichlet boundary control. The controllability to the trajectories of such a system was proved in the last decade by using Carleman estimates. Here, we go a step further by establishing the exact controllability in a space of analytic functions with the aid of the flatness approach.

The cost of boundary controllability for a parabolic equation with inverse square potential

The goal of this paper is to analyze the cost of boundary null controllability for the $ 1-D $ linear heat equation with the so-called inverse square potential: $ u_t -u_{xx} - \frac{\mu}{x^2} u = 0, \qquad x\in (0,1), \ t \in (0,T), $ where $ \mu $ is a real parameter such that $ \mu \leq 1/4 $. Since the works by Baras and Goldstein [4,5], it is known that such problems are well-posed for any $ \mu \leq 1/4 $ (the constant appearing in the Hardy inequality) whereas instantaneous blow-up may occur when $ \mu>1/4 $. For any $ \mu \leq 1/4 $, it has been proved in [52] (via Carleman estimates) that the equation can be controlled (in any time $ T>0 $) by a locally distributed control. Obviously, the same result holds true when one considers the case of a boundary control acting at $ x = 1 $. The goal of the present paper is to provide sharp estimates of the cost of the control in that case, analyzing its dependence with respect to the two paramaters $ T>0 $ and $ \mu \in (-\infty, 1/4] $. Our proofs are based on the moment method and very recent results on biorthogonal sequences.

Carleman Estimates of Some Stochastic Degenerate Parabolic Equations and Application

This paper is devoted to establishing global Carleman estimates for some forward and backward stochastic degenerate parabolic equations. First, two Carleman estimates for the forward equation are d...

Convexification for the Inversion of a Time Dependent Wave Front in a Heterogeneous Medium

An inverse scattering problem for the 3-dimensional acoustic equation in time domain is considered. The unknown spatially distributed speed of sound is the subject of the solution of this problem. A single location of the point source is used. Using a Carleman weight function, a globally strictly convex cost functional is constructed. A new Carleman estimate is proven. Global convergence of the gradient projection method is proven. Numerical experiments are conducted.

On an Inverse Source Problem for the Full Radiative Transfer Equation with Incomplete Data

A new numerical method to solve an inverse source problem for the radiative transfer equation involving the absorption and scattering terms, with incomplete data, is proposed. No restrictive assumption on those absorption and scattering coefficients is imposed. The original inverse source problem is reduced to boundary value problem for a system of coupled partial differential equations of the first order. The unknown source function is not a part of this system. Next, we write this system in the fully discrete form of finite differences. That discrete problem is solved via the quasi-reversibility method. We prove the existence and uniqueness of the regularized solution. Especially, we prove the convergence of regularized solutions to the exact one as the noise level in the data tends to zero via a new discrete Carleman estimate. Numerical simulations demonstrate good performance of this method even when the data are highly noisy.

Semiclassical resolvent estimates for bounded potentials

We study the cut-off resolvent of semiclassical Schr{\"o}dinger operators on $\mathbb{R}^d$ with bounded compactly supported potentials $V$. We prove that for real energies $\lambda^2$ in a compact interval in $\mathbb{R}_+$ and for any smooth cut-off function $\chi$ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has \begin{equation*} \| \chi (-h^2\Delta + V-\lambda^2)^{-1} \chi \|_{L^2\to H^1} \leq C \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This bound shows in particular an upper bound on the imaginary parts of the resonances $\lambda$, defined as a pole of the meromorphic continuation of the resolvent $(-h^2\Delta + V-\lambda^2)^{-1}$ as an operator $L^2_{\mathrm{comp}}\to H^2_{\mathrm{loc}}$: any resonance $\lambda$ with real part in a compact interval away from $0$ has imaginary part at most \begin{equation*} \mathrm{Im} \lambda \leq - C^{-1} \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This is related to a conjecture by Landis: The principal Carleman estimate in our proof provides as well a lower bound on the decay rate of $L^2$ solutions $u$ to $-\Delta u = Vu$ with $0\not\equiv V\in L^{\infty}(\mathbb{R}^d)$. We show that there exist a constant $M>0$ such that for any such $u$, for $R>0$ sufficiently large, one has \begin{equation*} \int_{B(0,R+1)\backslash \overline{B(0,R)}}|u(x)|^2 dx \geq M^{-1}R^{-4/3} \mathrm{e}^{-M \|V\|_{\infty}^{2/3} R^{4/3}}\|u\|^2_2. \end{equation*}