Global Carleman Estimate Research Articles

A new global Carleman estimate for the one-dimensional Kuramoto–Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem

In this paper, we establish a new global Carleman estimate for the linearized Kuramoto–Sivashinsky equation which is a one-parameter Carleman estimate. Based on this estimate, we can obtain the local exact controllability to the trajectories and Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation.

A quantitative internal unique continuation for stochastic parabolic equations

This paper is addressed to a quantitative internal unique continuation property for stochastic parabolic equations, i.e., we show that each of their solutions can be determined by the observation on any nonempty open subset of the whole region in which the equations evolve. The proof is based on a global Carleman estimate.

Inverse problems for the fourth order Schrödinger equation on a finite domain

In this paper we establish a global Carleman estimate for the fourth order Schrodinger equation with potential posed on a $1-d$ finite domain. The Carleman estimate is used to prove the Lipschitz stability for an inverse problem consisting in recovering a stationary potential in the Schrodinger equation from boundary measurements.

Stabilization of hyperbolic equations with mixed boundary conditions

This paper is devoted to study decay properties of solutions to hyperbolic equations in a bounded domain with two types of dissipative mechanisms, i.e. either with a small boundary or an internal damping. Both of the equations are equipped with the mixed boundary conditions. When the Geometric Control Condition on the dissipative region is not satisfied, we show that sufficiently smooth solutions to the equations decay logarithmically, under sharp regularity assumptions on the coefficients, the damping and the boundary of the domain involved in the equations. Our decay results rely on an analysis of the size of resolvent operators for hyperbolic equations on the imaginary axis. To derive this kind of resolvent estimates, we employ global Carleman estimates for elliptic equations with mixed boundary conditions.

Stability of an inverse problem for the discrete wave equation and convergence results

Using uniform global Carleman estimates for semi-discrete elliptic and hyperbolic equations, we study Lipschitz and logarithmic stability for the inverse problem of recovering a potential in a semi-discrete wave equation, discretized by finite differences in a 2-d uniform mesh, from boundary or internal measurements. The discrete stability results, when compared with their continuous counterparts, include new terms depending on the discretization parameter h. From these stability results, we design a numerical method to compute convergent approximations of the continuous potential.

Carleman estimate for a one-dimensional system of m coupled parabolic PDEs with BV diffusion coefficients

This paper is devoted to deriving global Carleman estimate for a one-dimensional linear coupled parabolic system of m equations with bounded variations (BV) diffusion coefficients. This kind of estimate is a generalization of the scalar result (Le Rousseau in J. Differ. Equ. 233:417-447, 2007). The key ingredient is to derive a global Carleman estimate for piecewise- diffusion coefficients based on the construction of a suitable weight function. The Carleman estimate in the case of BV diffusion coefficients is then obtained using the approach of BV diffusion coefficients by piecewise-constant coefficients. This Carleman estimate is used to show the observability inequality which yields the controllability result. MSC: 35K40, 26A45, 93B07.

Global Carleman estimate for stochastic parabolic equations, and its application

This paper is addressed to proving a new Carleman estimate for stochastic parabolic equations. Compared to the existing Carleman estimate in this respect (see [S. Tang and X. Zhang, SIAM J. Control Optim. 48 (2009) 2191–2216.], Thm. 5.2), one extra gradient term involving in that estimate is eliminated. Also, our improved Carleman estimate is established by virtue of the known Carleman estimate for deterministic parabolic equations. As its application, we prove the existence of insensitizing controls for backward stochastic parabolic equations. As usual, this insensitizing control problem can be reduced to a partial controllability problem for a suitable cascade system governed by a backward and a forward stochastic parabolic equation. In order to solve the latter controllability problem, we need to use our improved Carleman estimate to establish a suitable observability inequality for some linear cascade stochastic parabolic system, while the known Carleman estimate for forward stochastic parabolic equations seems not enough to derive the desired inequality.

A Carleman estimate for infinite cylindrical quantum domains and the application to inverse problems

We consider the inverse problem of determining the time-independent scalar potential q of the dynamic Schrödinger equation in an infinite cylindrical domain Ω, from one Neumann boundary observation of the solution. Assuming that q is known outside some fixed compact subset of Ω, we prove that q 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 Schrödinger operator acting in an unbounded cylindrical domain, the measurement of the Neumann data is performed on an infinitely extended subboundary of the cylinder.

Global Uniqueness for an Inverse Stochastic Hyperbolic Problem with Three Unknowns

This paper is addressed to an inverse stochastic hyperbolic problem with three unknowns, i.e., a random force intensity, an initial displacement, and an initial velocity. The global uniqueness for this inverse problem is proved by means of a new global Carleman estimate for the stochastic hyperbolic equation. It is found that both the formulation of stochastic inverse problems and the tools to solve them differ considerably from their deterministic counterpart. © 2015 Wiley Periodicals, Inc.

A NEW UNIQUE CONTINUATION PROPERTY FOR THE KORTEWEG–DE VRIES EQUATION

AbstractThe aim of this paper is to obtain a new unique continuation property (UCP) for the Korteweg–de Vries equation posed on a finite interval. Compared with the previous UCP, we need fewer conditions on the solution. For this purpose, we have to establish a global Carleman estimate for the Korteweg–de Vries equation.

Exact Controllability for Stochastic Transport Equations

This paper studies the exact controllability for stochastic transport equations by two controls: one is a boundary control imposed on the drift term and the other is an internal control imposed on the diffusion term. By means of the duality argument, this controllability problem can be reduced to an observability problem for backward stochastic transport equations, and the desired observability estimate is obtained by a new global Carleman estimate. Also, we present some results about the lack of exact controllability, which show that the action of two controls is necessary. To some extent, this indicates that the controllability problems for stochastic PDEs differ from their deterministic counterpart.

Insensitizing controls for the Cahn-Hilliard type equation

This paper is addressed to showing the existence of insensitizing controls for the one-dimensional Cahn-Hilliard type equation with a superlinear nonlinearity. We solve this problem by reducing the original problem to a controllability problem. The crucial point in this paper is an observability estimate for a linearized cascade system of the Cahn-Hilliard type equation. In order to obtain this observability estimate, we establish a global Carleman estimate for a fourth order parabolic operator.

Null controllability of retarded parabolic equations

We address in this work the null controllability problem for a linear heat equation with delay parameters. The control is exerted on a subdomain and we show how the global Carleman estimate due to Fursikov and Imanuvilov can be applied to derive results in this direction.

On the determination of the principal coefficient from boundary measurements in a KdV equation

Abstract This paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg–de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhgeĭm–Klibanov method.

Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation

In this article, we present an inverse problem for the nonlinear 1D Kuramoto–Sivashinsky (KS) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion 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 for this inverse problem is our main result and it relies on the Bukhgeĭm–Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.

Observability estimate and state observation problems for stochastic hyperbolic equations

This paper is devoted to a study of the boundary and internal state observation problems for stochastic hyperbolic equations. For this, we derive a boundary and an internal observability inequality for stochastic hyperbolic equations with nonsmooth lower order terms. The required inequalities are obtained by the global Carleman estimate for stochastic hyperbolic equations. By these inequalities, we get stability estimates for the state observation problems. As a consequence, we also establish a unique continuation property for stochastic hyperbolic equations.

Numerical controllability of the wave equation through primal methods and Carleman estimates

This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not apply in this work the usual duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null controls a functional involving weighted integrals of the state and the control. The optimality conditions show that both the optimal control and the associated state are expressed in terms of a new variable, the solution of a fourth-order elliptic problem defined in the space-time domain. We first prove that, for some specific weights determined by the global Carleman inequalities for the wave equation, this problem is well-posed. Then, in the framework of the finite element method, we introduce a family of finite-dimensional approximate control problems and we prove a strong convergence result. Numerical experiments confirm the analysis. We complete our study with several comments.

Exact controllability for stochastic Schrödinger equations

This paper is addressed to studying the exact controllability of stochastic Schrödinger equations by two controls. One is a boundary control and the other is an internal control in the diffusion term. By means of the duality argument, the control problem is converted into an observability problem for backward stochastic Schrödinger equations, and the desired observability estimate is obtained by a global Carleman estimate. At last, we give a result about the lack of exact controllability, which shows that the action of two controls is necessary.

Inverse problem for a free transport equation using Carleman estimates

This paper is devoted to prove a stability result for an absorption coefficient for a free transport equation in a smooth domain . The result is obtained using a global Carleman estimate with only one observation on a part of the boundary.

Carleman Estimates and null controllability of coupled degenerate systems

In this paper, we study the null controllability of weakly degenerateparabolic systems with two different diffusion coefficients and one control force.To obtain this aim, we had to develop new global Carleman estimates for adegenerate parabolic equation, with weight functions different from the ones of [2], [10] and [31].