Using $G^*$ as a navigation function

After the cost-to-go values stabilize, the resulting cost-to-go function, $G^*$ can be used as a navigation function. Even though $G^*$ is defined only over $S \subset X$, the value of the navigation function can be obtained using interpolation over any point in $R(S)$. The optimal action is selected as the one that satisfies the $\min$ in (8.6). This means that the state trajectory does not have to visit the grid points as in the Manhattan model. A trajectory can visit any point in $R(S)$, which enables trajectories to converge to the true optimal solution as $\Delta t$ and the grid spacing tend to zero.