Global Carleman Estimate Research Articles

Control of waves on Lorentzian manifolds with curvature bounds

We prove boundary controllability results for wave equations (with lower-order terms) on Lorentzian manifolds with time-dependent geometry satisfying suitable curvature bounds. The main ingredient is a novel global Carleman estimate on Lorentzian manifolds that is supported in the exterior of a null (or characteristic) cone, which leads to both an observability inequality and bounds for the corresponding constant. The Carleman estimate also yields a unique continuation result on the null cone exterior, which has applications toward inverse problems for linear waves on Lorentzian backgrounds.

Approximation theorem for the Kawahara operator and its application in the control theory

AbstractControl properties of the Kawahara equation are considered when the equation is posed on an unbounded domain. Precisely, the paper's main results are related to an approximation theorem that ensures the exact (internal) controllability in . Following [23], the problem is reduced to prove an approximate theorem which is achieved thanks to a global Carleman estimate for the Kawahara operator.

Null controllability for stochastic coupled systems of fourth order parabolic equations

This paper aims to establish null controllability for systems coupled by two backward fourth order stochastic parabolic equations. The main goal is to control both equations with only one control act on the drift term. To achieve this, we develop a new global Carleman estimate for fourth order stochastic parabolic equations, which allows us to deduce a suitable observability inequality for the adjoint systems.

An inverse problem for a transmission wave equation with a flat interface in Rn

In this paper we study a wave equation with discontinuous principal coefficient within a bounded domain of arbitrary dimension. It is obtained the stability of the inverse problem of recovering a space-dependent coefficient by observing a trace of the corresponding solution on part of the boundary. We provide a precise estimate of the minimum required time, as a function of the velocity change and domain size. The main tools are new global Carleman estimates for the transmission system with a particular weight function adapted to the interface geometry, which allows to obtain an optimal estimate of the minimum time.

Unique continuation for a fourth-order stochastic parabolic equation

In this paper, we prove a boundary unique continuation property for a fourth-order stochastic parabolic equation evolving in a domain G⊂Rn. Our result shows that the value of the solution can be determined by the observation on an arbitrary open subset of the boundary. The quantitative version of this property can be derived by the global Carleman estimate, which is deduced from a weighted identity for a fourth-order stochastic parabolic operator. The results in this paper are also new even if the fourth-order stochastic parabolic equation reduces to the corresponding fourth-order deterministic parabolic equation.

Global Lipschitz stability for inverse problems of wave equations on Lorentzian manifolds

We prove global Lipschitz stability for an inverse source problem of a system of wave equations on a Lorentzian manifold. The method used in this paper is widely known as the Bukhgeim–Klibanov method. However, the conventional method is not sufficient for the application to the hyperbolic partial differential equation with time-dependent coefficients to obtain the Lipschitz stability, and further innovations are needed. In this paper, we present an improved global Carleman estimate and an energy estimate to obtain the Lipschitz stability.

Null controllability for a structurally damped stochastic plate equation

In this paper, we obtain a null controllability result for a structurally damped refined stochastic plate equation with two kinds of boundary conditions. The key point is to establish an observability inequality for the backward structurally damped stochastic plate equation. For this purpose, we derive a new global Carleman estimate for the damped stochastic plate operator with homogeneous Dirichlet boundary conditions on the solution and the laplacian of the solution and with homogeneous Neumann boundary conditions of first and third order, respectively. Such an estimate is obtained by using the Carleman estimate for stochastic parabolic operators with complex coefficients in an iteration manner.

Carleman Estimates of Refined Stochastic Beam Equations and Applications

This paper is devoted to establishing global Carleman estimates for refined stochastic beam equations. First, by establishing a fundamental weighted identity, two Carleman estimates are derived with different weight functions for the refined stochastic beam equation, which is a coupled system consisting of a stochastic ordinary differential equation and a stochastic partial differential equation. As applications of these Carleman estimates, the exact controllability of the refined system is proved by the least controls in some sense. Different from classical stochastic beam equations, the refined one is exactly controllable at any time. Meanwhile, the uniqueness of an inverse source problem for refined stochastic beam equations is obtained without any requirement on the initial and terminal data.

Lipschitz stability in inverse source problems for degenerate/singular parabolic equations by the Carleman estimate

Abstract In this paper, we study the Lipschitz stability in inverse source problems for degenerate/singular parabolic equations in the case of a boundary observation. First, we establish new global Carleman estimates, which improve that derived by Vancostenoble (2011). Then, following the general lines of the approach introduced by Imanuvilov and Yamamoto (1998), we prove the Lipschitz stability in the inverse source problems. The results obtained are natural extensions of many previous results for parabolic equations without/with degeneracy or singularity.

An inverse problem for the transmission wave equation with Kelvin–Voigt damping

ABSTRACT In this paper, we consider a transmission wave problem with Kelvin–Voigt damping in two embedded domains in . We construct a global Carleman estimate for the transmission strongly damped wave equation with discontinuous coefficients. With the new Carleman estimate, we prove the Lipschitz stability and uniqueness of the inverse problem of determining the coefficient of the zeroth-order term in the equation with a single measurement observed in any small domain located in the outer domain.

Global Carleman estimate and its applications for a sixth-order equation related to thin solid films

<p style='text-indent:20px;'>Considered herein is the initial boundary value problem associated with a sixth-order nonlinear parabolic equation in a bounded domain. We first establish a new global Carleman estimate for the sixth-order parabolic operator. Based on this estimate, we obtain the local exact controllability to the trajectories and the unique continuation property of the parabolic equation.</p>

Inverse problem of reconstruction of degenerate diffusion coefficient in a parabolic equation

We consider the inverse problem of identification of degenerate diffusion coefficient of the form x α a(x) in a one dimensional parabolic equation by some extra data. We first prove by energy methods the uniqueness and Lipschitz stability results for the identification of a constant coefficient a and the power α by knowing interior data at some time. On the other hand, we obtain the uniqueness result for the identification of a general diffusion coefficients a(x) and also the power α form boundary data on one side of the space interval. The proof is based on global Carleman estimates for a hyperbolic problem and an inversion of the integral transform similar to the Laplace transform. Finally, the theoretical results are satisfactory verified by numerically experiments.

Inverse problems for first-order hyperbolic equations with time-dependent coefficients

We prove global Lipschitz stability for inverse source and coefficient problems for first-order linear hyperbolic equations, the coefficients of which depend on both space and time. We use a global Carleman estimate, and a crucial point, introduced in this paper, is the choice of the length of integral curves of a vector field generated by the principal part of the hyperbolic operator to construct a weight function for the Carleman estimate. These integral curves correspond to the characteristic curves in some cases.

Stabilization of the Weakly Coupled Wave-Plate System with One Internal Damping

We study a stabilization problem of a system coupled by a wave and a Euler–Bernoulli plate equation. Only one of the two equations is directly damped. Under some assumptions on the damping and the coupling terms, we prove that sufficiently smooth solutions of the system decay logarithmically without any geometric conditions on the damping domain. The proofs of these decay results rely on the interpolation inequalities for a coupled elliptic-parabolic system and the estimate of the resolvent operator for that system. The main tools to derive the desired interpolation inequalities are global Carleman estimates.

Local Exact Controllability to the Trajectories of Burgers–Fisher Equation

This paper is addressed to study local exact controllability to the trajectories of the Burgers–Fisher (BF) equation. By using the global Carleman estimate for the second-order parabolic operator, we establish the observable inequality and obtain the exact controllability to the trajectories of the linear system. Then, by local inverse theory, we consider the controllability result for the Burgers–Fisher equation.

Inverse source problem related to one-dimensional Saint-Venant equation

ABSTRACT The one-dimensional Saint-Venant equation describes unsteady water flow in channels and is derived from the one-dimensional Euler equation by imposing several physical assumptions. In this paper, we consider the linearized and simplified equation in the one-dimensional case featuring a mixed derivative term and prove the global Lipschitz stability of the inverse source problem via the global Carleman estimate.

Inverse problems for stochastic parabolic equations with additive noise

Abstract In this paper, we study two inverse problems for stochastic parabolic equations with additive noise. One is to determinate the history of a stochastic heat process and the random heat source simultaneously by the observation at the final time 𝑇. For this inverse problem, we obtain a conditional stability result. The other one is an inverse source problem to determine two kinds of sources simultaneously by the observation at the final time and on the lateral boundary. The main tool for solving the inverse problems is a new global Carleman estimate for the stochastic parabolic equation.

Local state observation for stochastic hyperbolic equations

In this paper, we solve a local state observation problem for stochastic hyperbolic equations without boundary conditions, which is reduced to a local unique continuation property for these equations. This result is proved by a global Carleman estimate. As far as we know, this is the first result in this topic.

Lipschitz stability for a semi-linear inverse stochastic transport problem

Abstract This paper is addressed to a semi-linear stochastic transport equation with three unknown parameters. It is proved that the initial displacement, the terminal state and the random term in diffusion are uniquely determined by the state on partial boundary and a Lipschitz stability of the inverse problem is established. The main tool we employ is a global Carleman estimate for stochastic transport equations.

Exact controllability for the semilinear Mindlin-Timoshenko system

We consider the dynamical one-dimensional Mindlin–Timoshenko system for beams. We obtain a global exact controllability result for this semilinear system with superlinear nonlinearities. For this purpose, we establish an observability estimate for the linearized system with bounded potentials. Moreover, we obtain an explicit estimate of the observability constant in terms of the norms of the potentials. Such an estimate is obtained by a combination of a pointwise estimate, a global Carleman estimate for hyperbolic differential operators and an analysis of the regularity of solutions of a auxiliary optimal control problem. The controllability of the semilinear system is proved using the Hilbert Uniqueness Method (HUM) and a fixed point technique.