Gradient constraint for unbounded consumption ($V_{\xi }-V_w\le 0$)

If consumption contributes $c(V_{\xi }-V_w)$ to the Hamilton-Jacobi-Bellman equation and has no finite upper bound, finite value requires $V_{\xi }\le V_w$. Strict inequality makes zero consumption optimal locally. Equality describes an active transfer boundary, often reached through singular consumption control.