 Threshold energy scheduling with a completion reward

For energy remaining $x$, $\dot{x}=-u$, $0\le u\le 1$, running cost $up(t)$ and terminal cost $-R(\tau )$, the normal maximizing Hamiltonian is $u(p_{\ast }-p(t))$, where the constant costate defines $p_{\ast }$. Hence full power is chosen below the threshold and zero power above it. Free completion time on an active terminal arc gives $p_{\ast }+R^{\prime }(\tau )=p(\tau )$. Global optimality still requires a comparison of feasible completion times; the maximum principle is only a necessary condition.