Runge–Kutta tableaux and stage collocation equations

Let $s\ge 1$ be a stage count, $A=(a_{ik})\in {\mathbb{R}}^{s\times s}$, and $b,c\in {\mathbb{R}}^s$ satisfy $c_i={\sum }_{k=1}^s{a}_{ik}$ for each $i$. This is a (possibly implicit) Runge–Kutta tableau. On mesh cell $[t_j,t_{j+1}]$ with width $h_j$, a start state $X_{j,0}$ and piecewise-constant control $u_j$, define stage times $t_{j,i}=t_j+c_i{h}_j$ and the collocation stage equations $X_{j,i}=X_{j,0}+h_j{\sum }_{k=1}^s{a}_{ik}f(t_{j,k},X_{j,k},u_j),i=1,\dots ,s\mathrm{.}$ The equations are admissibly defined only for $X_{j,0}\in {\mathrm{\Omega }}_x$, $u_j\in {\mathrm{\Omega }}_u$, and stage inputs $(t_{j,i},X_{j,i},u_j)\in [0,T]\times {\mathrm{\Omega }}_x\times {\mathrm{\Omega }}_u$ for every $i$, using the domains in Finite-horizon ODE optimal-control transcription data. A transcription procedure must check these memberships before each query of $f$ or $L$, including trial stages of an implicit solve, and report a stage-time-domain failure for an invalid time or a stage-domain failure for an invalid state or control without making that query. The corresponding endpoint equation is $X_{j+1,0}=X_{j,0}+h_j{\sum }_{i=1}^s{b}_{i}f(t_{j,i},X_{j,i},u_j)$. In a finite transcription, an implicit tableau's coupled stage residuals are solved internally and the stage variables are eliminated from the outer NLP; the procedure returns the nonlinear stage-solve status and residuals. The quadrature contribution of the cell is $h_j{\sum }_{i=1}^s{b}_{i}L(t_{j,i},X_{j,i},u_j)$. This definition asserts no stage-solution existence, uniqueness, order, stability, or error bound. If $A$ is strictly lower triangular, evaluate stages in increasing order $i=1,\dots ,s$: form $X_{j,i}=X_{j,0}+h_j{\sum }_{k<i}{a}_{ik}{F}_{j,k}$, check the stated time-state-control memberships, and then evaluate $F_{j,i}=f(t_{j,i},X_{j,i},u_j)$, reporting an unavailable value as a stage-evaluation failure. This uses the explicit tableau coefficients and algebraic recursion of Explicit Runge–Kutta tableaus, stage maps, and failure status, with the finite-horizon domain checks specified here; it does not invoke that item's open-time-domain stage map or require an extension of $f$ beyond $[0,T]$.