Approximating the cost functional

A discrete cost functional must be derived from the continuous cost functional, (8.39). The final term is just assigned as $l_F(x_F) = l_F(x(t_f))$. The cost at each stage is

$$l_d(x_k,u_k) = \int_0^{\Delta t} l(x(t),u(t)) dt,$$ (8.65)

and $l_d(x_k,u_k)$ is used in the place of $l(x_k,u_k)$ in (8.56). For many problems, the integral does not need to be computed repeatedly. To obtain Euclidean shortest paths, $l_d(x_k,u_k) = \Vert u_k\Vert$ can be safely assigned for all $x_k \in X$ and $u_k \in U(x_k)$. A reasonable approximation to (8.65) if $\Delta t$ is small is $l(x(t),u(t)) \Delta t$.