Identification of time active limit with lower and upper bounds of total amount loaded by unknown sources in 2D transport equations

Abstract

We address the nonlinear inverse source problem, which consists in identifying the time limit from which some unknown sources in a two-dimensional advection–dispersion-reaction equation become inactive. Under some reasonable assumptions and without requiring any a priori information about the form of the involved unknown sources, we prove the identifiability of the desired time active limit. We establish an identification method based on some boundary null controllability results that uses the records of the associated state on the outflow boundary and of its flux on the inflow boundary of the monitored domain to determine the time active limit. The application of this method to the most encountered forms of surface water pollution sources leads us to determine also lower and upper bounds that provide a significant approximation of the total amount loaded by the involved unknown sources without requiring any a priori information either on the number of active sources or on the form of their time-dependent intensity functions. Some numerical experiments on a variant of the surface water biological oxygen demand pollution model are presented.