A file of $X$ gigabytes (GB) is to be transmitted over a communications link. At each time $t$ the sender can choose a transmission rate $u(t)$ within the range $[0,1]$ GB 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 $t=0$. 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 $x(T)=0$. Using Pontryagin's maximum principle, or otherwise, 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 $X=1$. Show that it is optimal to transmit at a constant rate $u(t)=1$ between times $T-1 \leqslant t \leqslant T$, where $T$ is the unique positive solution to the equation

$$\frac{1}{(T-1) T}=2 \gamma T+1$$