Identification for control: Transversal flows and collinearity in the inverse conductivity problem

Time independent monophasic flow in a confined, saturated porous medium is modelled by an elliptic equation. Let Omega subset of R(n), n = 2, 3, be a bounded, convex domain. The inverse problem consists of identifying position dependent conductivity, a(x), x in Omega, from interior measurements. On its turn, a(.) satisfies a stationary transport equation. Two uniqueness conditions are presented herewith, based on transversal gradient flows and on Ito - Kunisch collinearity, respectively. The former implies an invertible coordinate transformation and simplifies the application of the method of characteristics; the latter provides a first integral of the defect equation for the inverse problem. Both of them have geophysical motivation. The stability of the map F^{-1}, which takes hydraulic potential data to a(.), is estimated in W^{1,p} (Omega), where p depends on the applicable uniqueness condition. The coefficient a(.) appears in subsequent direct (control) problems, which yield the simulation result, r, and which are described by the map G. Control - oriented identification requires i) to estimate the stability of the data-to-output map, G . F^{-1}, and ii) to provide computable expressions for the estimation constants. Both requirements are met herewith. Constants are determined up to the norm of a raising up function. The stability of G . F^{-1} improves in a special case, where another collinearity condition provides a first integral in the control problem.