Ober-Blöbaum, Junge & Marsden (2008/2011) prove the discrete costate recursion converges to the continuous Pontryagin adjoint as step size → 0, under stated smoothness/coercivity/discretization hypotheses

pontryagincostateadjointconvergencesymplectic-integratorsmarsdenhageroptimal-controldiscretizationkelley-bryson

The positive case for discrete-to-continuous adjoint convergence is not folklore — it is a proven theorem, with stated hypotheses. Ober-Blöbaum, Junge & Marsden (2008 preprint / 2011 ESAIM: COCV) show that for Lagrangian optimal-control problems meeting a Smoothness assumption and a Coercivity assumption (ensuring a locally unique, Lipschitz-continuous control-from-costate map), the discrete adjoint/costate recursion generated by a symplectic partitioned Runge-Kutta discretization of the state equations converges to the continuous costate system as the step size h → 0, at matching order: "If the symplectic partitioned Runge-Kutta discretization of the state system is of order κ and bi > 0 for each i, then the scheme for the adjoint system is again a symplectic partitioned Runge-Kutta scheme of the same order" (their Theorem 3.15), via a general consistency-plus-stability convergence theorem credited to Hager (2001) and Dontchev, Hager & Veliov (2000).

This is the proven half of claim-kelley-bryson-pontryagin-costate-same-object-only-under-hypotheses's conditional: the discrete Kelley-Bryson costate recursion genuinely does converge to Pontryagin's continuous adjoint ODE — inside a hypothesis set (smoothness, coercivity, structure-preserving discretization) that a generic feedforward network with ReLU activations and an arbitrary forward-Euler step does not automatically satisfy, per claim-dto-and-otd-neural-ode-gradients-can-diverge.