State transversality conditions that can be used with Pontryagin's maximum principle and say when they are helpful.

Given $T$, it is desired to maximize $c_{1} x_{1}(T)+c_{2} x_{2}(T)$, where

$$\begin{aligned} &\dot{x}_{1}=u_{1}\left(a_{1} x_{1}+a_{2} x_{2}\right), \\ &\dot{x}_{2}=u_{2}\left(a_{1} x_{1}+a_{2} x_{2}\right), \end{aligned}$$

and $u=\left(u_{1}, u_{2}\right)$ is a time-varying control such that $u_{1} \geqslant 0, u_{2} \geqslant 0$ and $u_{1}+u_{2}=1$. Suppose that $x_{1}(0)$ and $x_{2}(0)$ are positive, and that $0<a_{2}<a_{1}$ and $0<c_{1}<c_{2}$. Find the optimal control at times close to $T$. Show that over $[0, T]$ the optimal control is constant, or makes exactly one switch, the latter happening if and only if

$$c_{2} e^{a_{2} T}<c_{1}+\frac{a_{1} c_{2}}{a_{2}}\left(e^{a_{2} T}-1\right)$$