Localization of unknown sources for parabolic systems on the basis of available observations

Abstract

The paper considers two problems. Firstly the nonlinear localization problem; on the basis of available observations on the diffusion process, recover the location of an unknown single source generating this process. In general, the solution of this problem is set-valued and disconnected. Secondly, the identifiability problem; what types of observations (finite-dimensional at every moment of time, as generally occurs in physical situations) are able to provide enough information to restore the location point? A new approach is given, based on the introduction of a space of test-functions: in order to determine the unknown location, one needs to analyse a relevant system of algebraic equations. The latter can be determined in advance (‘off-line’). Sufficient conditions of identifiability are derived and the duality relations between the above nonlinear localization problems and the associated adjoint linear control problems are established. The above problems are motivated by environmental monitoring.