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This review examines classical and recent results on controllability and inverse problems for hyperbolic and dispersive equations with dynamic boundary conditions. We aim to illustrate the applicability of Carleman estimates to establish…
We propose a new numerical method for the solution of the problem of the reconstruction of the initial condition of a quasilinear parabolic equation from the measurements of both Dirichlet and Neumann data on the boundary of a bounded…
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…
We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequencies as the data. A conditional Lipschitz stability estimate for the inverse problem holds in the case of…
We establish Lipschitz stability for both the potential and the initial conditions from a single boundary measurement in the context of a hyperbolic boundary initial value problem. In our setting, the initial conditions are allowed to…
In this paper, we investigate an inverse Cauchy problem for a stochastic hyperbolic equation. A Lipschitz type observability estimate is established using a pointwise Carleman identity. By minimizing the constructed Tikhonov-type…
We consider the linear system of viscoelasticity with the homogeneous Dirichlet boundary condition. First we prove a Carleman estimate with boundary values of solutions of viscoelasticity system. Since a solution $u$ under consideration is…
In this article, we improve the classical Bukhgeim-Klibanov method presented in [1],which can be used to prove the conditional stability of inverse source problem for a hyperbolic equation from the measurement on the subboundary. A major…
We consider the first and half order time fractional equation with the zero initial condition. We investigate an inverse source problem of determining the time-independent source factor by the data at an arbitrarily fixed time and we…
In this paper, we investigate an inverse random source problem concerned with recovering the strength of a random, uncorrelated acoustic source from correlation measurements of emitted time-harmonic acoustic waves. Such problems arise in…
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…
A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problems with the initial condition the delta function concentrated at a single plane (i.e. the plane…
In this paper we establish a global Carleman estimate for the fourth order Schr\"odinger equation posed on a $1-d$ finite domain. The Carleman estimate is used to prove the Lipschitz stability for an inverse problem consisting in retrieving…
The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed,…
We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic.…
We consider backward problems for semilinear coupled parabolic systems in bounded domains. We prove conditional stability estimates for linear and semilinear systems of strongly coupled parabolic equations involving general semilinearities.…
We consider the inverse hyperbolic problem of recovering all spatial dependent coefficients, which are the wave speed, the damping coefficient, potential coefficient and gradient coefficient, in a second-order hyperbolic equation defined on…
We consider the inverse problem of determining the initial states or the source term of a hyperbolic equation damped by some non-local time-fractional derivative. This framework is relevant to medical imaging such as thermoacoustic or…
We consider the inverse problem for the general transport equation with external field, source term and absorption coefficient. We show that the source and the absorption coefficients can be uniquely reconstructed from the boundary…
In this paper, we study the inverse problem for determining an unknown time-dependent source coefficient in a semilinear pseudo-parabolic equation with variable coefficients and Neumann boundary condition. This unknown source term is…