Carleman Estimates Research Articles

Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky system

<p style='text-indent:20px;'>In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- and second-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a relaxed observability inequality which yields a <inline-formula><tex-math id="M1">\begin{document}$ \phi(\triangle t) $\end{document}</tex-math></inline-formula>-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter <inline-formula><tex-math id="M2">\begin{document}$ \triangle t $\end{document}</tex-math></inline-formula> goes to 0. The proof relies on a known Carleman estimate for second-order time-discrete parabolic operators and a new Carleman estimate for the time-discrete fourth-order equation.</p>

Controllability results for cascade systems of m coupled N-dimensional stokes and Navier-stokes systems by N – 1 scalar controls

In this paper we deal with the controllability properties of a system of m coupled Stokes systems or m coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control a m × N state (corresponding to the velocities of the fluids) by N — 1 distributed scalar controls. The proof of the controllability of the coupled Stokes systems is based on a Carleman estimate for the adjoint system. The local null-controllability of the coupled Navier-Stokes systems is then obtained by means of the source term method and a Banach fixed point.

The Carleman-based contraction principle to reconstruct the potential of nonlinear hyperbolic equations

We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear Cauchy problems whose corresponding solutions converge to a function that can be used to efficiently compute an approximate solution to the inverse problem of interest. The convergence analysis is established by combining the contraction principle and Carleman estimates. We numerically solve the linear Cauchy problems using a quasi-reversibility method. Numerical examples are presented to illustrate the efficiency of the method.

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.

A Hölder stability estimate for a 3D coefficient inverse problem for a hyperbolic equation with a plane wave

Abstract A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problem with the initial condition being the delta function concentrated at a single plane (i.e. the plane wave). A certain associated operator is written in finite differences with respect to two out of three spatial variables, i.e. “partial finite differences”. The grid step size is bounded from below by a fixed number. A Carleman estimate is applied to obtain, for the first time, a Hölder stability estimate for this problem. Another new result is an estimate from below the amplitude of the first term of the expansion of the solution of the forward problem near the characteristic wedge.

A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications

We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented.

Point sources and stability for an inverse problem for a hyperbolic PDE with space and time dependent coefficients

We study stability aspects for the determination of space and time-dependent lower order perturbations of the wave operator in three space dimensions with point sources. The problems under consideration here are formally determined and we establish Lipschitz stability results for these problems. The main tool in our analysis is a modified version of Bukgheĭm-Klibanov method based on Carleman estimates.

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.

Internal Controllability of Parabolic Systems with Star- and Tree-like Couplings

We consider systems of parabolic equations coupled in zero order terms in a star-like or a tree-like shape, with an internal control acting in only one of the equations. We obtain local exact controllability to the stationary solutions of the system, under hypotheses concerning the supports of the coupling functions. The key point is establishing Carleman estimates with appropriate observation operators for the adjoint to the linearized system, which allows the study of the controllability problem, in either linear or nonlinear cases, in an $L^\infty$ framework.

Simultaneous recoveries for semilinear parabolic systems

In this paper, we study inverse boundary problems associated with semilinear parabolic systems in several scenarios where both the nonlinearities and the initial data can be unknown. We establish several simultaneous recovery results showing that the passive or active boundary Dirichlet-to-Neumann operators can uniquely recover both of the unknowns, even stably in a certain case. It turns out that the nonlinearities play a critical role in deriving these recovery results. If the nonlinear term belongs to a general C 1 class but fulfilling a certain growth condition, the recovery results are established by the control approach via Carleman estimates. If the nonlinear term belongs to an analytic class, the recovery results are established through successive linearization in combination with special complex geometrical optics solutions for the parabolic system.

Forecasting stock options prices via the solution of an ill-posed problem for the Black–Scholes equation

In the previous paper (2016 Inverse Problems 32 015010), a new heuristic mathematical model was proposed for accurate forecasting of prices of stock options for 1–2 trading days ahead of the present one. This new technique uses the Black–Scholes equation supplied by new intervals for the underlying stock and new initial and boundary conditions for option prices. The Black–Scholes equation was solved in the positive direction of the time variable, this ill-posed initial boundary value problem was solved by the so-called quasi-reversibility method (QRM). This approach with an added trading strategy was tested on the market data for 368 stock options and good forecasting results were demonstrated. In the current paper, we use the geometric Brownian motion to provide an explanation of that effectivity using computationally simulated data for European call options. We also provide a convergence analysis for QRM. The key tool of that analysis is a Carleman estimate.

Stable determination of coefficients in semilinear parabolic system with dynamic boundary conditions

In this work, we study the stable determination of four space-dependent coefficients appearing in a coupled semilinear parabolic system with variable diffusion matrices subject to dynamic boundary conditions which couple intern-boundary phenomena. We prove a Lipschitz stability result for interior and boundary potentials by means of only one observation component, localized in any arbitrary open subset of the physical domain. The proof mainly relies on some new Carleman estimates for dynamic boundary conditions of surface diffusion type.

Carleman estimate for complex second order elliptic operators with discontinuous Lipschitz coefficients

In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing the result by M. Bellassoued and J. Le Rousseau (2018) and the arguments by M. Di Cristo, E. Francini, C.-L. Lin, S. Vessella, and J.-N. Wang (2017), we present an elementary method to derive the Carleman estimate under the optimal regularity assumption on the coefficients.

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.

Null Controllability for a Degenerate Population Equation with Memory

In this paper we consider the null controllability for a population model depending on time, on space and on age. Moreover, the diffusion coefficient degenerate at the boundary of the space domain. The novelty of this paper is that for the first time we consider the presence of a memory term, which makes the computations more difficult. However, under a suitable condition on the kernel we deduce a null controllability result for the original problem via new Carleman estimates for the adjoint problem associated to a suitable nonhomogeneous parabolic equation.

Stability of a coefficient inverse problem for the discrete Schrödinger equation and a convergence result

In this paper, we prove the stability and convergence of a coefficient inverse problem for the semi-discrete Schrödinger equation by means of an appropriate Carleman estimate. First, we establish a new discrete Carleman estimate for the semi-discrete Schrödinger equation discretized by finite differences. Then by the developed Carleman estimate, we prove the stability of the semi-discrete Schrödinger equation. The consistency between the semi-discrete inverse problem and the continuous one is also proved. With the consistency and stability, the convergence of the semi-discrete inverse problem can be obtained based on the Lax equivalence theorem.

Observability Estimate for the Wave Equation with Variable Coefficients

This paper is devoted to a study of observability estimate for the wave equation with variable coefficients $(h^{jk}(x))_{n\times n}$ ($n\in\mathbb{N})$. We consider both the observation point lies outside the domain and the observation point lies inside the domain. Based on a Carleman estimate for the ultra-hyperbolic operator and a delicate treatment of observation region, we obtain two observability estimates with explicit observability constants. The key improvements are: (1) we improve the requirement of waiting time $T$; (2) we improve the size of the observation region (see Fingure 1 and Fingure 2 for the case of $(h^{jk}(x))_{n\times n}=I_n)$ .

Global uniqueness in an inverse problem for a class of damped stochastic plate equations

This paper deals with the global uniqueness of an inverse problem for stochastic plate equations with structural damping. The key point is the Carleman estimate for the fourth‐order stochastic plate operators . For this aim, a weighted pointwise identity for a fourth‐order stochastic plate operator is established, via which we obtained the desired Carleman estimate for the corresponding stochastic plate equation with structural damping.

A Gauge-Invariant Unique Continuation Criterion for Waves in Asymptotically Anti-de Sitter Spacetimes

We reconsider the unique continuation property for a general class of tensorial Klein–Gordon equations of the form □gϕ+σϕ=G(ϕ,∇ϕ),σ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Box _{g} \\phi + \\sigma \\phi = {\\mathcal {G}}(\\phi ,\ abla \\phi ) \ ext {,} \\qquad \\sigma \\in {\\mathbb {R}} \\end{aligned}$$\\end{document}on a large class of asymptotically anti-de-Sitter spacetimes. In particular, we aim to generalize the previous results of Holzegel, McGill, and the second author (Holzegel and Shao in Commun Math Phys 347(3):723–775, 2016; Commun Partial Differ Equ 42(12):1871–1922, 2017; McGill and Shao in Class Quantum Gravity 38(5):054001, 2021) (which established the above-mentioned unique continuation property through novel Carleman estimates near the conformal boundary) in the following ways: We replace the so-called null convexity criterion—the key geometric assumption on the conformal boundary needed in McGill and Shao (2021) to establish the unique continuation properties—by a more general criterion that is also gauge invariant.Our new unique continuation property can be applied from a larger, more general class of domains on the conformal boundary.Similar to McGill and Shao (2021), we connect the failure of our generalized criterion to the existence of certain null geodesics near the conformal boundary. These geodesics are closely related to the classical Alinhac-Baouendi counterexamples to unique continuation (Alinhac and Baouendi in Math Z 220(4):561–568, 1995). Finally, our gauge-invariant criterion and Carleman estimate will constitute a key ingredient in proving unique continuation results for the full nonlinear Einstein-vacuum equations, which will be addressed in a forthcoming paper of Holzegel and the second author (Holzegel and Shao in Unique continuation for the Einstein equations in asymptotically anti-de sitter spacetimes (in preparation), 2022).

Interior control of waves on time dependent domains

We obtain a novel interior control result for wave equations on time dependent domains. This is done by deriving a suitable Carleman estimate and proving the corresponding observability inequality. We consider the wave equation with time dependent lower order coefficients, without any time analyticity assumptions. Moreover, we obtain improved control regions when compared with standard Carleman methods.