Identification of source terms in heat equation with dynamic boundary conditions

Abstract

We study an inverse parabolic problem of identifying two source terms in heat equation with dynamic boundary conditions from a final time overdetermination data. Using a weak solution approach by Hasanov, the associated cost functional is analyzed, especially a gradient formula of the functional is proved and given in terms of the solution of an adjoint problem. Next, the existence and uniqueness of a quasi-solution are also investigated. Finally, the numerical reconstruction of some heat sources in a 1-D equation is presented to show the efficiency of the proposed algorithm.

Logarithmic convexity and impulsive controllability for the 1-D heat equation with dynamic boundary conditions

“…where ψ Γ = ψ |Γ is the trace of ψ, ∆ Γ is the Laplace-Beltrami operator and ∂ ν ψ is the normal derivative with respect to the outward unit normal vector field ν. The controllability and inverse problems for the non-impulsive heat equation with the dynamic boundary condition (1.2) have been considered in the recent papers [3,5,9,20,28]. The impulse approximate controllability has been recently investigated in [13].…”

Identification of source terms in wave equation with dynamic boundary conditions

“…• Although we have only considered a simplified model of hyperbolic systems with dynamic boundary conditions, our approach can be generalized to more general models as where Ω ⊂ R N (N ≤ 3) is a bounded domain with smooth boundary Γ, a ∈ L ∞ (Ω), b ∈ L ∞ (Γ), and d, γ > 0 are given speed constants. • Comparing to the heat equation with dynamic boundary conditions, the proposed Landweber scheme in [12] becomes slow for the wave equation (1.1). This issue can be interpreted in terms of the measured data we have considered.…”

“…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.…”

Impulse null approximate controllability for heat equation with dynamic boundary conditions

“…We mention the papers [15,32], which physically derive the dynamic boundary conditions. Furthermore, the controllability and inverse problems of heat equation with dynamic boundary conditions have recently been investigated in [2,3,7,19,25], where the authors have proven controllability and stability results by proving new Carleman estimates.…”