Bounds for Detectability of Material Damage by Noisy Electrical Measurements

The problem of detectability of inclusions by boundary measurements is considered. The inclusions diier from the background material by their conductivity, the location of inclusions is unknown. The detectability is equal to the diierence of energies needed to inject currents in the model body without inclusion and in the sample under investigation. We consider a game where inclusions hide in the domain and current distribution seeks them; this is either max-min or a min-max problem for detectability. These problems have diierent solutions and diierent physical meaning, both are solved and the bounds of detectability are established. The bounds are independent of the shape of the domain and have a simple analytical form. DESCRIPTION OF THE PROBLEM We deal with the problem of detection of a damage by nondestructive boundary electrical measurements. We answer the question: whether or not a conducting body contains damaged regions inside. We are not interested in nding the location of these regions, their shape, etc., only in detecting the possible presence of damage. A damaged part of the sample is called inclusion; it is supposed that the conductivity I of inclusions is constant and is less than constant conductivity B of a basic material. The problem of detection naturally arises in technical diagnostics, where one is interested in detecting cracks, corrosion, or any other defect in a sample.