Determination of an unknown source for a thermoelastic system with a memory effect

Abstract

We study an inverse problem of determining a spatially varying source term in a thermoelastic medium with a memory effect. The coupling phenomena between elasticity and heat as well as the memory effect make such an inverse problem very complicated. We firstly prove a pointwise Carleman estimate for a general strongly coupled hyperbolic system, and then obtain a Carleman estimate for the hyperbolic thermoelastic system. Based on this estimate, we finally establish a Hölder stability for the inverse source problem only by making a displacement measurement on a given subdomain for sufficiently large times, provided the source be known near the boundary. The uniqueness for such an inverse problem is yielded as a direct result.

Recovery of a space-dependent vector source in thermoelastic systems

“…Using a Carleman estimate, a Hölder stability for the inverse source problem is proved, which implies the uniqueness of the inverse source problem. Wu and Liu [10] studied an inverse source problem of determining p(x) for type-II thermoelasticity, i.e. k = 0 and ρ = 0.…”

Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems

“…Wu and Liu [148] considered the MCIP for the thermoelastic system with memory (also, see their preceding work [147]). That system is…”

“…Let ω ⊂ Ω be a subdomain. The MCIP in [148] consists in the recovery of the source term p (x), given the vector function f (x, t) = u (x, t) for (x, t) ∈ ω × (0, T ) . The function σ (x, t) is assumed to be known.…”

“…The function σ (x, t) is assumed to be known. Lipschitz stability estimate for this MCIP was obtained in [148].…”

Ionic parameters identification of an inverse problem of strongly coupled PDE’s system in cardiac electrophysiology using Carleman estimates

“…However, to our best knowledge, if the matrix (a i,j ) 1≤i,j≤2 formed by the parameters a 1,1 , a 2,2 describing the self diffusion of the state variables and a 1,2 , a 2,1 describing the influence of each component on another, is diagonalizable then the transformed system satisfies a weakly coupled system. In this case, Carleman estimate is obtained similarly to that of the weakly coupled system, see [8,21,50]. But due to the fact that the anisotropy in the bidomain equations depends on the fiber directions, and the fact that the fiber direction are space dependent, one cannot diagonalize the problem as in [8,50].…”