A file of $X \mathrm{Mb}$ is to be transmitted over a communications link. At each time $t$ the sender can choose a transmission rate, $u(t)$, within the range $[0,1]$ Mb per second. The charge for transmitting at rate $u(t)$ at time $t$ is $u(t) p(t)$. The function $p$ is fully known at time 0. If it takes a total time $T$ to transmit the file then there is a delay cost of $\gamma T^{2}$, $\gamma>0$. Thus $u$ and $T$ are to be chosen to minimize

$$\int_{0}^{T} u(t) p(t) d t+\gamma T^{2}$$

where $u(t) \in[0,1], d x(t) / d t=-u(t), x(0)=X$ and $x(T)=0$. Quoting and applying appropriate results of Pontryagin's maximum principle show that a property of the optimal policy is that there exists $p^{*}$ such that $u(t)=1$ if $p(t)<p^{*}$ and $u(t)=0$ if $p(t)>p^{*}$.

Show that the optimal $p^{*}$ and $T$ are related by $p^{*}=p(T)+2 \gamma T$.

Suppose $p(t)=t+1 / t$ and $X=1$. For what value of $\gamma$ is it optimal to transmit at a constant rate 1 between times $1 / 2$ and $3 / 2$ ?