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ABSTRACT This paper concerns the null controllability of stochastic degenerate parabolic equation with convection term in weakly degenerate case. Due to the degeneracy, we first transfer to study an approximate nondegenerate system. Next, we establish the Carleman estimate for backward stochastic degenerate parabolic equation with convection term. In order to deal with the convection term, the Carleman estimate is established by introducing an auxiliary function to transfer the diffusion and the convection term into a whole complex divergence‐like form. By this Carleman estimate, we then obtain an observability inequality for the adjoint system of the approximate system. Based on the observability inequality and an approximate argument, we proved our null controllability result. [ABSTRACT FROM AUTHOR]

The goal of this article is the analyses of inverse source problem and null controllability for a heat equation with multipolar inverse-square potentials: $ \small {\sum _{j=1}^m}\frac {\mu _j}{|x-x^j|^2}, $ ∑ j = 1 m μ j | x − x j | 2 , where $ x^1, x^2,\ldots,x^m $ x 1 , x 2 , ... , x m are different points, defined in a bounded $ C^2 $ C 2 domain $ \Omega \subset \mathbb R^n $ Ω ⊂ R n , and $ \mu _1,\ldots,\mu _m $ μ 1 , ... , μ m are real constants. More precisely, we show that for any $ \mu _j\leq \frac {(n-2)^2}{4},\ j=1,\ldots,m, $ μ j ≤ (n − 2) 2 4 , j = 1 , ... , m , we can obtain a Lipschitz stability result in inverse source problems and we will show that the system is null controllable. Our proofs are mainly based on Carleman estimates, and due to multiple singularities, we give a new Carleman estimate by choosing an appropriate weight function. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the inverse problem for the Moore-Gibson-Thompson equation with a memory term and variable diffusivity, which introduce a sort of delay in the dynamics, producing nonlocal effects in time. The Hölder stability of simultaneously determining the spatially varying viscosity coefficient and the source term is obtained by means of the key pointwise Carleman estimate for the Moore-Gibson-Thompson equation. For the sake of generality in mathematical tools, the analysis of this paper is discussed within the framework of Riemannian geometry. [ABSTRACT FROM AUTHOR]

This paper is about Hölder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time t=T$$ \left\{t=T\right\} $$. In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem. [ABSTRACT FROM AUTHOR]

For an inverse coefficient problem of determining a state‐varying factor in the corresponding Hamiltonian for a mean field game system, we prove the global Lipschitz stability by spatial data of one component and interior data in an arbitrarily chosen subdomain over a time interval. The proof is based on Carleman estimates with different norms. [ABSTRACT FROM AUTHOR]

The primary focus of this paper is to establish the internal null controllability for the one-dimensional heat equation featuring dynamic boundary conditions. This achievement is realized by introducing a new Carleman estimate and an observability inequality for the corresponding backward system. In conclusion, the paper includes a set of numerical experiments that serve to confirm the validity of the theoretical findings and underscore the effectiveness of the designed control with a minimal |$L^{2}$| -norm. [ABSTRACT FROM AUTHOR]

Control 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 (0,+∞)$(0,+\infty)$. Following [23], the problem is reduced to prove an approximate theorem which is achieved thanks to a global Carleman estimate for the Kawahara operator. [ABSTRACT FROM AUTHOR]

In this paper we establish a Lebeau--Robbiano spectral inequality for a degenerate one-dimensional elliptic operator, with an optimal dependency with the frequency parameter. The proof relies on a combination of uniform local Carleman estimates away from the degeneracy and a moment method adapted for a degenerate elliptic operator. We also provide an application to the null-controllability on a measurable set in time for the associated degenerate heat equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with an inverse problem of a coupled thermoelastic plate model. Two major features are that the thermal equation has a memory effect, while the plate equation has a curved middle surface. A differential geometric approach is developed, by which we study the pointwise Carleman estimates for elliptic and hyperbolic equations. We are able to prove a key Carleman estimate for the strongly coupled system. From them, the Hölder stability in recovering the source terms and the coupling coefficient is obtained. The measurements of the plate deflection and temperature are assumed to be taken on a subdomain of the boundary. [ABSTRACT FROM AUTHOR]

The inverse problem of reconstructing two space-varying coefficients in a system of one-dimensional (1-d) time-dependent advection-diffusion-reaction (ADR) equations is considered. The ADR system can be used as a water quality model which describes the evolution of the biochemical oxygen demand (BOD) and dissolved oxygen (DO) in a river or stream. The coefficients to be reconstructed represents the effect of the deoxygenation and superficial reaeration processes on the DO and BOD concentration in water. Hölder stability estimates for the coefficients of interest are established using the Carleman estimate technique. [ABSTRACT FROM AUTHOR]

In a bounded domain $ \Omega \subset \mathbb{R}^d $, $ d\geq 1 $, over a time interval $ (0,T) $, we consider mean field game equations whose principal coefficients depend on the time and the state variables with a general Hamiltonian. We attach a non-zero Robin boundary condition. We first prove the Lipschitz stability in $ \Omega \times (\varepsilon, T- \varepsilon) $ with given $ \varepsilon>0 $ for the determination of the solutions by the associated Dirichlet data on an arbitrarily chosen subboundary of $ \partial \Omega $. Next we prove the Lipschitz stability for an inverse problem of determining spatially varying factors of source terms and a coefficient by extra boundary data and spatial data at an intermediate time. [ABSTRACT FROM AUTHOR]

We study an inverse problem of determining a time-dependent damping coefficient and potential appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines the time-dependent damping coefficient and potential uniquely. [ABSTRACT FROM AUTHOR]

We study an inverse problem for variable coefficient fractional parabolic operators of the form (∂t - div(A(x)▽x))s + q(x, t) for s ∈ (0, 1) and show the unique recovery of q from exterior measured data. Similar to the fractional elliptic case, we use a Runge-type approximation argument, which is obtained via a global weak unique continuation property. The proof of such a unique continuation result involves a new Carleman estimate for the associated variable coefficient extension operator. In the latter part of the work, we prove analogous unique determination results for fractional parabolic operators with drift. [ABSTRACT FROM AUTHOR]

The aim of this paper is to survey some recent works on inverse problems for stochastic partial differential equations (SPDEs for short), which were solved by means of Carleman estimate. For simplicity, we focus on two typical SPDEs, i.e, stochastic hyperbolic equations and stochastic parabolic equations. Our particular concerns are those inverse problems which are genuinely stochastic and therefore cannot be reduced to any deterministic ones. Also, some open problems are presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the energy decay of a coupled wave system with one local Kelvin–Voigt damping. The system consists of two wave equations locally coupled by zero‐order terms. When the damping coefficient function and the coupling coefficient function satisfy suitable assumptions, by Carleman estimate and the frequency‐domain method, we show that the energy of the system decays logarithmically for sufficiently smooth initial data. [ABSTRACT FROM AUTHOR]

We propose a new globally convergent numerical method to compute solution Hamilton-Jacobi equations defined in R d , d ≥ 1 , on a truncated bounded domain. This method is named the Carleman convexification method. By Carleman convexification, we mean that we use a Carleman weight function to convexify the conventional least squares mismatch functional. We will prove a new version of the convexification theorem guaranteeing that the mismatch functional involving the Carleman weight function is strictly convex and, therefore, has a unique minimizer. Moreover, a consequence of our convexification theorem guarantees that the minimizer of the Carleman weighted mismatch functional is an approximation of the viscosity solution we want to compute. Some numerical results in 1D and 2D will be presented. [ABSTRACT FROM AUTHOR]

Using an appropriate Carleman-type estimate, we establish a result of unique continuation for a special class of one-dimensional systems that model the evolution of long water waves with small amplitude in the presence of surface tension. [ABSTRACT FROM AUTHOR]

In this paper, we study the null controllability of systems of $ n $-coupled parabolic equations with dynamic boundary conditions, where the coupling and control matrices $ A $ and $ B $ are constant in time and space. Being different to the case of static boundary conditions, we will show that the Kalman rank condition $ rank[B, AB,\dots, A^{n-1}B ] = n $ is a sufficient condition, we also show that it is necessary for the null controlability under an extra assumption on the boundary coupling. The null controlability result will be proved by proving Carleman and observability inequalities for the corresponding adjoint problem. [ABSTRACT FROM AUTHOR]

The purpose of this work is to investigate the exponential stability of a second‐order coupled wave equations by Laplacian with one locally internal viscous damping. The strong stability is established by combining a unique continuation result and a Carleman estimate. Besides, the exponential stability is proved under (PMGC) condition on the damping region without any restriction on wave propagation speed (i.e., whether the two wave equations propagate at the same speed or not). The proof of this result relies on the frequency domain method combined with the multiplier techniques. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the cost of null controllability for a backward stochastic degenerate parabolic equation with a boundary control in the vanishing viscosity limit. Firstly we obtain a new Carleman estimate for the adjoint stochastic equation. Combining this Carleman estimate and an exponential dissipation estimate, we obtain the uniform null controllability for large control time. When the waiting time tends to 0, we prove that the cost of null controllability increases to infinity. [ABSTRACT FROM AUTHOR]

We are concerned with the null controllability of a linear coupled population dynamics system or the so-called prey-predator model with Holling type I functional response of predator wherein both equations are structured in age and space. It is worth mentioning that in our case, the space variable is viewed as the "gene type" of population. The studied system is with two different dispersion coefficients which depend on the gene type variable and degenerate in the boundary. This system will be governed by one control force. To reach our goal, we develop first a Carleman type inequality for its adjoint system and consequently the pertinent observability inequality. Note that such a system is obtained via the original paradigm using the Lagrangian method. Afterwards, with the help of a cost function we will be able to deduce the existence of a control acting on a subset of the gene type domain and which steers both populations of a certain class of age to extinction in a finite time.\looseness-2

In this paper, we obtain the exact controllability for a refined stochastic wave equation with three controls by establishing a novel Carleman estimate for a backward hyperbolic-like operator. Compared with the known result [Q. L\"u and X Zhang, Mathematical Control Theory for Stochastic Partial Differential Equations, Springer, Cham, Switzerland, 2021], the novelty of this paper is twofold: (1) Our model contains the effects in the drift terms when we put controls directly in the diffusion terms, which is more sensible for practical applications; (2) We provide an explicit description of the waiting time which is sharp in the case of dimension one and is independent of the coefficents of lower terms. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering problem involving the timedomain elastic wave equations in a bounded d-dimensional domain. First, an explicit formula for the density reconstruction is established by means of the Dirichlet-to-Neumann operator. The reconstruction is mainly based on the modified boundary control method and complex geometric optics solutions for the elastic wave. Next, the stable observability is obtained by a Carleman estimate. Finally, the stability for the density is presented by the connect operator. [ABSTRACT FROM AUTHOR]

We consider, for h,E>0, the semiclassical Schrödinger operator -h²Δ+V-E in dimension two and higher. The potential V, and its radial derivative $\dell_{r}V$ are bounded away from the origin, have long-range decay and V is bounded by r-δ near the origin while $\dell_{r}V$ is bounded by r-1-δ, where 0≤δ≤4(2-√-1). In this setting, we show that the resolvent bound is exponential in h-1, while the exterior resolvent bound is linear in h-1. [ABSTRACT FROM AUTHOR]

Longtime behavior of the Kirchhoff plate equations with locally distributed internal damping is studied. The equation is equipped with either homogeneous Dirichlet boundary conditions of zero order and second order, or Robin boundary conditions of first order and third order. Under some assumptions on the damping, it is shown that sufficiently smooth solutions of the equation decay logarithmically. [ABSTRACT FROM AUTHOR]

In this paper, we establish a result of unique continuation for a special two-dimensional nonlinear system that models the evolution of long water waves with small amplitude in the presence of surface tension. More precisely, we will show that if (η , Φ) = (η (x , y , t) , Φ (x , y , t)) is a solution of the nonlinear system, in a suitable function space, and (η , Φ) vanishes on an open subset Ω of R 2 × [ - T , T ] , then (η , Φ) ≡ 0 in the horizontal component of Ω. To state such property, we use a Carleman-type estimate for a differential operator L related to the system. We prove the Carleman estimate using a particular version of the well known Treves’ inequality. [ABSTRACT FROM AUTHOR]

In this paper, we prove a global Carleman estimate for the complex Ginzburg-Landau operator with a cubic nonlinear term in a bounded domain of Rn, n = 2, 3. As applications, we study state observation problems for the Ginzburg-Landau equation. [ABSTRACT FROM AUTHOR]

This paper concerns the null controllability of an one-dimensional degenerate parabolic equation with a slightly superlinear reaction term in a bounded interval, with a control on a subinterval. The degeneracy of the equation is relatively weak, and the reaction term grows relatively slow at infinity. It is noted that the behavior of the reaction term at infinity is consistent with the well-known null controllability results for heat equations. The Carleman estimate and the observability inequality for the corresponding linear conjugate equations are established. Subsequently, the control problem is proved to be null controllable by using the observability inequality, the the Kakutani fixed point theorem and some a priori estimates. [ABSTRACT FROM AUTHOR]

In this note, we establish a new Carleman estimate with singular weights for the sub-Laplacian on a Carnot group G for functions satisfying the discrepancy assumption in (2.16) below. We use such an estimate to derive a sharp vanishing order estimate for solutions to stationary Schrödinger equations. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying two multiobjective problems for stochastic degenerate parabolic equations. The first one is a hierarchical control problem, in which the controls are classified into a pair of leaders and a pair of followers. For each pair of leaders, a Nash equilibrium is searched for a noncooperative game problem. The aim of the pair of leaders is to achieve null controllability of the system. The other multiobjective problem is an inverse initial problem under a Nash equilibrium strategy. In contrast to the classical inverse initial problem, an optimization problem for stochastic degenerate parabolic equations is first investigated. Then, the conditional stability of determining initial information is derived through terminal observation. [ABSTRACT FROM AUTHOR]

The first globally convergent numerical method for a coefficient inverse problem for the Riemannian radiative transfer equation (RRTE) is constructed. This is a version of the so-called convexification method, which has been pursued by this research group for a number of years for some other CIPs for PDEs. Those PDEs are significantly different from the RRTE. The presence of the Carleman weight function in the numerical scheme is the key element which insures the global convergence. Convergence analysis is presented along with the results of numerical experiments, which confirm the theory. RRTE governs the propagation of photons in the diffuse medium in the case when they propagate along geodesic lines between their collisions. Geodesic lines are generated by the spatially variable dielectric constant of the medium. [ABSTRACT FROM AUTHOR]

We are concerned with the null controllability of a linear coupled population dynamics system or the so-called prey-predator model with Holling type I functional response of predator wherein both equations are structured in age and space. It is worth mentioning that in our case, the space variable is viewed as the “gene type” of population. The studied system is with two different dispersion coefficients which depend on the gene type variable and degenerate in the boundary. This system will be governed by one control force. To reach our goal, we develop first a Carleman type inequality for its adjoint system and consequently the pertinent observability inequality. Note that such a system is obtained via the original paradigm using the Lagrangian method. Afterwards, with the help of a cost function we will be able to deduce the existence of a control acting on a subset of the gene type domain and which steers both populations of a certain class of age to extinction in a finite time. [ABSTRACT FROM AUTHOR]

An overdetermination is introduced in an initial condition for the second order mean field games system (MFGS). This makes the resulting problem close to the classical ill-posed Cauchy problems for PDEs. Indeed, in such a problem an overdetermination in boundary conditions usually takes place. A Lipschitz stability estimate is obtained. This estimate implies uniqueness. A new Carleman estimate is derived. This latter estimate is called "quasi-Carleman estimate", since it contains two test functions rather than a single one in conventional Carleman estimates. These two estimates play the key role. Carleman estimates were not applied to the MFGS prior to the recent work of Klibanov and Averboukh in [M. V. Klibanov and Y. Averboukh, Lipschitz stability estimate and uniqueness in the retrospective analysis for the mean field games system via two Carleman estimates, preprint 2023, https://arxiv.org/abs/2302.10709]. [ABSTRACT FROM AUTHOR]

We consider solutions satisfying the Neumann zero boundary condition and a linearized mean field game system in $ \Omega \times (0, T) $, where $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ and $ (0, T) $ is the time interval. We prove two kinds of stability results in determining the solutions. The first is Hölder stability in time interval $ (\varepsilon, T) $ with arbitrarily fixed $ \varepsilon>0 $ by data of solutions in $ \Omega \times \{T\} $. The second is the Lipschitz stability in $ \Omega \times (\varepsilon, T- \varepsilon) $ by data of solutions in arbitrarily given subdomain of $ \Omega $ over $ (0, T) $. [ABSTRACT FROM AUTHOR]

We consider the dynamic elasticity equation, modeled by the Euler Bernoulli plate equation, with a locally distributed singular structural (or viscoelastic) damping in a boundary domain. Using a frequency domain method based on the Batty and Duyckaerts result [7], combined with an estimate of Carleman type we provide precise decay estimate showing that the energy of the system decays logarithmically as the type goes to the infinity. [ABSTRACT FROM AUTHOR]

For linearized Navier–Stokes equations, we consider an inverse source problem of determining a spatially varying divergence-free factor. We prove the global Lipschitz stability by interior data over a time interval and velocity field at t 0 > 0 over the spatial domain. The key machinery are Carleman estimates for the Navier–Stokes equations and the operator rot. [ABSTRACT FROM AUTHOR]

We consider stability in an inverse problem of determining the material coefficient matrix for a coupled system that describes acoustic interactions, by the Riemannian geometrical approach. The stability is proved by the Carleman estimates and observability inequalities. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the boundary controllability of heat equation with dynamic boundary conditions. More precisely, we prove that the equation is null controllable at any positive time by means of a boundary control supported on an arbitrary subboundary. The proof of the main result combines a new boundary Carleman estimate and some regularity estimates for the adjoint system, with an explicit dependence with respect to the final time. This technique allows us to overcome a new difficulty that arises when absorbing a normal derivative term. [ABSTRACT FROM AUTHOR]

In this paper, we retrieve a stationary potential in the strongly coupled Schrödinger equations from either boundary or internal measurements. Lipschitz stabilities are obtained by means of new Carleman estimates for the system with different observations, respectively. [ABSTRACT FROM AUTHOR]

Motivated by the application to helioseismology, we demonstrate uniqueness and stability for a class of inverse problems of the weighted transport equation. Using 퐴-analytic functions, this inverse problem is expressed as a Cauchy problem. In this form, we show that, for a finite even trigonometric polynomial weight function, the resulting system is well-conditioned numerically and permits a Carleman-like estimate with boundary terms. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 dyt−ρΔytdt+Δ2ydt$$ d{y}_t-\rho \Delta {y}_t dt+{\Delta}^2 ydt $$. 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. [ABSTRACT FROM AUTHOR]

We study the problem of stabilization for the acoustic system with a spatially dis- tributed damping. Without imposing any hypotheses on the structural properties of the damping term, we identify logarithmic decay of solutions with growing time. Logarithmic decay rate is shown through a new Carleman estimate which allows us to carry out a special analysis for the resolvent. [ABSTRACT FROM AUTHOR]

This paper is devoted to a sharp internal/boundary observability inequality for a hyperbolic system with a zero order potential. For this purpose, we first establish a new Carleman estimate for hyperbolic operator in H1-norm. Based on this Carleman estimate and a modified auxiliary optimal control problem, we obtain Carleman estimate for hyperbolic operator in L2-norm. Then, by virtue of a modified energy estimate and a delicate treatment of the observation region, we obtain an internal observability estimate with the observability constant of the order exp (C‖q‖2/3L∞(Q;ℝN×N)), with q the potential involved in the system. We also address the same problem for boundary observation. Compared with the related results in the literature, the main contributions of this paper are the observability constant is sharper, the waiting time T is shorter and the internal (or boundary) observation domain is smaller. [ABSTRACT FROM AUTHOR]

This paper is addressed to establishing controllability and observability for some forward linear stochastic complex degenerate/singular Ginzburg-Landau equations. It is sufficient to establish appropriate observability inequalities for the corresponding backward and forward equations. The key is to prove the Carleman estimates of the forward and backward linear stochastic complex degenerate/singular Ginzburg-Landau operators. Compared with the existing deterministic results, it is necessary to overcome the difficulties caused by some complex coefficients and random terms. The results obtained cover those of deterministic cases and generalize those of stochastic degenerate parabolic equations. Moreover, the limit behavior of the coefficients in the equation is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a transmission wave equation in N embedded domains with multiple interfaces of discontinuous coefficients in ℝ2. We study the global stability in determining the source term from a one-measurement data of wavefield velocity in a subboundary over a time interval. We prove the stability estimate for this inverse source problem by a combination of the local hyperbolic/elliptic Carleman estimates and the Fourier-Bros-Iagolniter transformation. Our method could be generalized to general dimensions since the weight functions and Carleman estimates are independent of the dimensions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]