A direct, unconditional discrete-time analogue of Pontryagin's maximum principle does not exist in general — the maximization condition can fail without added convexity

pontryaginmaximum-principleoptimal-controldiscrete-timeconvexitykelley-brysonhalkin

Before neural-ODE training raised the discrete-vs-continuous adjoint question, control theory had already disputed and resolved it negatively in the general case. Kipka & Gupta (2017/2019, SIAM J. Control Optim.) summarize: "the formulation of the maximum principle for discrete-time optimal control problems becomes quite tricky since the maximization condition cannot be expected to hold in general and in fact some of the early literature on this topic was mathematically incorrect... However, the maximization condition does hold under appropriate convexity assumptions on the dynamics and moreover an approximate maximization condition can be derived (under suitable assumptions) in the absence of such assumptions."

The specific claim this establishes: it is the maximization (Hamiltonian-optimality) condition of Pontryagin's apparatus — not the costate/adjoint equation alone — that fails to carry over to discrete time without extra convexity assumptions the continuous theory does not need. For the counterexamples, Kipka & Gupta's own citation — "(see [6, 27] for a number of interesting counterexamples)" — points to Boltyanskii's Optimal Control of Discrete Systems (1978) and Mordukhovich's Variational Analysis and Generalized Differentiation II (2006). The attribution of the original counterexamples to Hubert Halkin's 1966 SIAM J. Control paper rests on web secondary sources only — Halkin appears nowhere in Kipka & Gupta's reference list — as does the credit to Butkovskiy for first showing a direct discrete extension is impossible in general; none of these primary texts has been independently read.

This is the distinction claim-dto-and-otd-neural-ode-gradients-can-diverge rediscovers inside neural-ODE training: the maximization condition and the costate recursion are not the same object, and only the latter has a clean convergence theorem (claim-ober-blobaum-junge-marsden-prove-discrete-adjoint-converges-to-continuous).