Source Localization Problem for Parabolic Systems

Two problems for the linear distributed parameter systems of parabolic type motivated by environmental monitoring are discussed: (1) Nonlinear localization problem: recover the location of an unknown single source on the basis of available observations. In general, the solution of this problem is set-valued and disconnected. (2) Identifiability problem: what types of observations are able to ensure enough information to restore the location point? An approach is given, based on the introduction of a suitable space of test-functions: in order to determine the unknown location, one has to analyze a proper system of algebraic equations. The latter can be constructed in advance. Sufficient conditions for identifiability are derived and the duality relations between the above nonlinear problems and the problems of open loop control and controllability for an associated adjoint linear system are established