Stability Estimates for Composite Identification-and-Control Maps Related to a Distributed Parameter System

Consider a distributed parameter system in one spatial dimension, governed by the ordinary linear differential equation $( a u ’ ) ’ =_{a.e.} f$, where $a$ is conductivity, $u$ the potential and $f$ the source term. Said equation models several time - independent physical processes of interest. Assume that measured {potential, source term} pairs, the \textit{data}, are available and that the eventual goal is not just to identify conductivity but to use the model to predict the potential, given new \textit{controls} i.e., boundary conditions and source terms. A composite identification-and-control map relates measured to predicted potentials via conductivity. The map is shown herewith to be Lipschitz continuous, provided the affected Banach spaces are suitably chosen and regularization applied. For the control map to be well-posed, conductivity has to exist and be uniquely determined from the data. If conductivity is regarded as the unknown in the equation above, a unique solution may follow from either a regular or a singular Cauchy problem. Both of these situations can be met in practice. They are compared herewith and are shown to affect the composite map. In particular, uniform $(W^{1, \infty})$ stability estimates for the predicted potentials are obtained from the regular Cauchy problem, whereas the singular one typically yields $(W^{1, 1})$-estimates for the same quantities.