Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions

Abstract

In this paper, we study an inverse problem for linear parabolic system with variable diffusion coefficients subject to dynamic boundary conditions. We prove a global Lipschitz stability for the inverse problem involving a simultaneous recovery of two source terms from a single measurement and interior observations, based on a recent Carleman estimate for such problems.

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

“…We have the following Carleman estimates, see [1] and [9] for the proof. (ii) If the functions F and F Γ are not necessarily zero, then there exist λ 2 ≥ 1, s 2 ≥ 1 and C 2 = C 2 (ω, Ω) > 0 such that the solution Q = (q, q Γ ) to (27) satisfies…”

“…This type of boundary conditions has been considered in the context of null controllability, when only one control is acting on the system, in [31,37]. The cost of approximate controllability and an inverse source problem of such equations were also studied in [9] and [1], respectively. For some controllability results of hyperbolic equation with dynamic boundary condition, the reader can see [22].…”

“…) is the solution to (1). For a fixed f ∈ L 2 (0, T ; L 2 (ω)), the pair (v 1 , v 2 ) = (v 1 (f ), v 2 (f )) is called a Nash equilibrium for (J 1 , J 2 ) associated to f if…”

Identification of source terms in heat equation with dynamic boundary conditions

“…Such an estimate was introduced, for the first time, in the study of multidimensional inverse problems by Bukhgeim and Klibanov, 10 and became a powerful tool to establish uniqueness and stability results. Recently in Ait Ben Hassi et al, 14 the authors have applied this method in the study of an inverse source problem (ISP) for general parabolic equations with dynamic boundary conditions from interior measurements. They established a Lipschitz stability estimate for the source terms.…”

Time and norm optimal controls for the heat equation with dynamical boundary conditions

“…For some first results on the controllability issue, we refer to Khoutaibi and Maniar 23 . The cost of approximate controllability and an inverse source problem of a general model of () were studied in Boutaayamou et al 24 and Ait Ben Hassi et al, 25 respectively.…”