Positive-interpolation semi-Lagrangian update for first-order HJB equations

Choose integers $N_j\ge 3$, a number $M\ge 1$ of time steps, and $\tau =T\mathrm{/}M$. Sample $u_0$ at the grid nodes of Periodic Cartesian grids and tensor-product multilinear interpolation and set $U_i^0=u_0(x_i)$. At each $n=0,\dots ,M-1$, compute $U_i^{n+1}={\min }_{a\in A_k}\left\{I_h{U}^n\right(x_i+\tau b_a(x_i,t_n))+\tau {\mathrm{\ell }}_a(x_i,t_n)\},t_n=n\tau \mathrm{.}$ The position is evaluated on the torus, so its coordinates wrap modulo one. The algorithm always uses a finite nonempty set $A_k\subset A$. For a finite control model take $A_k=A$; for a compact control model, supply finite sets whose Hamiltonian errors ${\eta }_k(R)$ from Periodic first-order controlled Hamilton–Jacobi Cauchy problem tend to zero on every bounded momentum set. This is the declared finite-control approximation branch; it does not presume an unimplemented continuous minimization oracle. Return $U^M$.

The update is the explicit Euler-footpoint semi-Lagrangian branch specified by the plan. It defines its footpoint and interpolant locally; no unauthored characteristic-flow estimate is presumed.