Monotonicity, L-infinity stability, and consistency of the semi-Lagrangian HJB step

If $V\le W$, [F2] gives $I_{h}V\le I_{h}W$ at every footpoint. Adding the same running cost preserves the inequality for each action, and taking the minimum preserves it again; hence $S_{h,\tau }V\le S_{h,\tau }W$. Also $\mathrm{\mid }{I}_{h}V\mathrm{\mid }\le \mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$ by the convex weights in [F2]. Every action value in [F1] therefore has absolute value at most $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}+\tau L$, and so does their minimum.

Let $y_a=x_i+\tau b_a(x_i,t_n)$ in a periodic lift. By [F2], $I_h\varphi (t_n,\cdot )(y_a)=\varphi (t_n,y_a)+O(h_{\max }^{}\mathrm{\parallel }{D}^2\varphi {\mathrm{\parallel }}_{\mathrm{\infty }})\mathrm{.}$ Taylor expansion at $(x_i,t_n)$ gives $\varphi (t_n,y_a)+\tau {\mathrm{\ell }}_a=\varphi (t_n,x_i)+\tau (b_a\cdot D\varphi +{\mathrm{\ell }}_a)(x_i,t_n)+O({\tau }^2{B}^2\mathrm{\parallel }{D}^2\varphi {\mathrm{\parallel }}_{\mathrm{\infty }})\mathrm{.}$ The minimum of finitely many entries changes by at most their largest perturbation. Replacing $A_k$ by $A$ changes the minimum of $b_a\cdot D\varphi +{\mathrm{\ell }}_a$ by at most ${\eta }_k(P)$, by [F3]. A time Taylor expansion then yields the normalized residual in the statement, because $-{\min }_{a\in A}(b_a\cdot D\varphi +{\mathrm{\ell }}_a)=H$.

Steps 1.1 and 1.2 prove order preservation, the stated sup-norm bound, and the local consistency error. The interpolation term is divided by $\tau$, so convergence of this residual requires the explicit condition $h_{\max }^{}\mathrm{/}\tau \to 0$. No solution-error rate follows from these one-step estimates.