Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology

Abstract

In this paper, we study the stability result for the conductivities diffusion coefficients to a strongly reaction-diffusion system modeling electrical activity in the heart. To study the problem, we establish a Carleman estimate for our system. The proof is based on the combination of a Carleman estimate and certain weight energy estimates for parabolic systems.where f and g are stimulation currents applied to Ω. We complete this model with Dirichlet boundary conditions for the intra-and extracellular electric potentials

Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component

“…Carleman estimate for n-coupled degenerate equations. Now, we show the main result of this section, which is the ω-Carleman estimate for the coupled system (1). For this purpose, the parameters γ, ρ and d will be chosen such that…”

“…where(y 0 k ) 1≤k≤n ∈ L 2 (0, 1) n , T > 0 fixed, Q := (0, T ) × (0, 1), the coupling terms b kj = b kj (t, x) ∈ L ∞ (Q) (1 ≤ k, j ≤ n) and the function a is a diffusion coefficient which degenerates at 0 (i.e., a(0) = 0) and which can be either weakly degenerate (WD), i.e., a ∈ C([0, 1]) ∩ C 1 ((0, 1]), a(0) = 0, a > 0 in (0, 1], ∃α ∈ [0, 1), such that xa (x) ≤ αa(x), ∀x ∈ [0, 1], (2) or strongly degenerate (SD), i.e.,…”

“…The essence of their methodology comes from the Bukhgeim-Klibanov method using the Carleman estimate to derive the global uniqueness in inverse problems, see [12]. Inverse problems for parabolic systems of two equations have also been studied, see for example [1,7,16,17,18,19,24,28]. There are, of course, other kinds of models which involve coupled systems of partial differential equations (wave-wave, hyperbolicparabolic coupled systems).…”

“…As a consequence, we prove a stability estimate of Lipschitz type in determining all the source terms from the knowledge of some measurements of only one component of the solution. To our knowledge, this paper is the first one concerning Lipschitz stability results in inverse problems for degenerate coupled systems such as (1).…”

“…Global Carleman estimates. In this section we give a new global Carleman estimate for the system (1). To this end, as in [15], we introduce the following time and space weight functions…”