An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions

Abstract

We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain.

Impulse null approximate controllability for heat equation with dynamic boundary conditions

“…As an application of the inequality (2), we study the null approximate impulse controllability of the heat equation with dynamic boundary conditions (1), where the control function acts on a subdomain ω and at one point of time τ ∈ (0, T ), (see more about impulse control in [26,34]). Definition 4.1 (see [37]).…”

“…The observability estimate for parabolic equations was initiated independently by [22] and [14] in the context of null controllability, based on Carleman inequalities (see also [12]). Furthermore, the observability estimate at one point of time, which is an estimate of the energy at a fixed time on the whole domain in terms of the energy at the same time but on a small subdomain, was already established for a linear heat equation with homogeneous Dirichlet boundary conditions by Phung in [28], using a new strategy combining the logarithmic convexity method and a Carleman commutator approach (see [1,30]). From the underlying observation at one time, many applications were derived as impulse approximate controllablity [29,31], which consists on finding an impulsive control steering approximately the system from an arbitrary initial state to a final state in a finite interval of time.…”

Identification of source terms in heat equation with dynamic boundary conditions

“…The first integral in the right-hand side of Equation ( 16) is null by the surface divergence formula (2). Adding up the two integrals (15) and (17), we obtain…”

“…They established a Lipschitz stability estimate for the source terms. In Ait Ben Hassi et al, 15 we have considered an inverse problem of radiative potentials and initial temperatures for the same equation. We have proven a Lipschitz stability estimate for the potentials and then obtained a logarithmic stability result for the initial conditions by a logarithmic convexity method.…”

Identification of source terms in wave equation with dynamic boundary conditions

“…In contrast to the large literature for static boundary conditions, there are not sufficient researches on inverse hyperbolic problems incorporating dynamic boundary conditions, in spite of the well-established literature for the direct problems. Some recent works have been lunched for inverse parabolic problems with dynamic boundary conditions [11][12][13]. As for direct problems, various theoretical approaches have been developed for the analysis of hyperbolic evolution equations with dynamic boundary conditions.…”