Finite-horizon ODE optimal-control transcription data

For pointwise numerical transcription on $[0,T]$, the data consist of integers $n,m\ge 1$, open sets ${\mathrm{\Omega }}_x\subseteq {\mathbb{R}}^n$ and ${\mathrm{\Omega }}_u\subseteq {\mathbb{R}}^m$, a nonempty compact convex control set $U_c\subset {\mathrm{\Omega }}_u$, an initial state $x_0\in {\mathrm{\Omega }}_x$, a pointwise-defined vector field $f:[0,T]\times {\mathrm{\Omega }}_x\times {\mathrm{\Omega }}_u\to {\mathbb{R}}^n$, a pointwise-defined running cost $L:[0,T]\times {\mathrm{\Omega }}_x\times {\mathrm{\Omega }}_u\to \mathbb{R}$, and a terminal cost $\varphi :{\mathrm{\Omega }}_x\to \mathbb{R}$. The admissible state solves $\dot{x}(t)=f(t,x(t),u(t))$ almost everywhere and the continuous objective is ${\int }_0^{T}L(t,x(t),u(t))dt+\varphi (x(T))$. Optional path and endpoint constraints are pointwise-defined functions evaluated at declared continuous times or transcription nodes. A compact tube is a compact subset of $[0,T]\times {\mathrm{\Omega }}_x\times {\mathrm{\Omega }}_u$ containing the state/control triples used by stages and trajectories. On such a tube, $f,L,\varphi$ and any constraint functions used in a local NLP theorem are continuous and have continuous state/control derivatives through order three (the $C^3$ convention of $C^k$ Euclidean maps and diffeomorphisms), with those derivatives bounded. An $L^{\mathrm{\infty }}$ equivalence class alone has no specified value at a Runge–Kutta stage time, so it is not an admissible stage-evaluation input without an additional pointwise representative and sampling rule. This definition makes no existence, convergence, or continuous-path-feasibility claim.