u^*(\tau -1) > u^*(\tau) \quad&\Longrightarrow\quad b^*(\tau) = B \qquad

(\textnormal{full battery)}

\end{align}

then $u^*(t)$ is optimal with respect to maximizing the minimal used energy among all

use functions and maximizes the utility $U(\tau, T)$.

\bigskip

Since the perfect use function $u(t)$ was chosen with knowledge of all harvested energy for all times, the battery level will drop to 0\% only at times, when the harvesting functions starts delivering more energy than we currently use. \\

The energy consumption will then be chosen, such that the battery is full, when the harvesting function is decreasing, thus after the battery is fully charged, the use function should decrease. \\