Formalize outcome-level equivalence of stochastic feedback controllers

Formalize an outcome-level equivalence relation for regime-indexed stochastic quadratic feedback controllers over declared subsets of times and coordinates, and derive a corresponding minimal-complexity selection criterion among certified candidate controllers.

Background

The paper observes that controllers can be substantially different as column-stochastic matrices while producing similar or jointly admissible localization trajectories on a specified set of times and coordinates. This extends the paper’s broader distinction between coefficient identifiability and predictive or decision-level equivalence: different operator realizations may be indistinguishable for the declared outcome even when their matrix coefficients differ.

The unresolved task is to turn this observation into a formal outcome-level equivalence notion and then use it to select a controller of minimal complexity among candidates that satisfy the required acceptability certification. Such a result would help avoid treating structurally distinct but functionally equivalent feedback policies as meaningfully different solutions.

References

More generally, two controllers $K_\nu,K_\nu'$ that are far apart as column-stochastic matrices may induce similar, or jointly admissible, localization trajectories on a declared subset $\mathcal T$ of times and coordinates. This disassociation between distance in policy and distance in outcome, already noted for constant allocations in Sec.~II.C, extends naturally to $\Pi_K{(s)}$; formalizing an outcome-level equivalence over $\mathcal T$ and a corresponding minimal-complexity selection criterion among certified candidates is left as an open direction.

— Structured Stochastic Representations of Integrated Dynamic Strategies  (2609.10998 - Vides, 10 Sep 2026) in Section II.D, subsection “Stochastic feedback closure and certified feedback feasibility”