Weighing operator perturbation from quasi-critical source system response

In Hilbert space, a linear source-to-flux problem in the critical (zero eigenvalue) limit is ill-posed, but regularized by a constraint on a linear functional, fulfilled by tuning some control variable. For any exciting perturbation, I obtain, by spectral decomposition and perturbation theory, the regularized flux and the regularizing control variable non-linear responses. May the exciting perturbation be obtained, inversely, from observable responses? Yes, in some cases, from the existence of a weight scale, a perturbation series, determined by recursion relations, involving well-posed source problems, and the possibility of obtaining this weight scale from observables of both the unconstrained and constrained systems.