I strive to prove that the following system of differential equations:

$$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$

has a unique Caratheodory solution on a given interval $[0,T]$, where $u:[0,T]\to [0,1]$ is a control, lets say measurable or continuous if necessary. I cannot apply Caratheodory existence theorem https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem since the function $f:[0,T]\times\mathbb{R}^2\to\mathbb{R}^2,\ f(t,x,y)=(x-u(t)xy,-y+u(t)xy)$ is not satisfying the Lipschitz-like condition, and the linearity-like growing in $(x,y)$. Someone told me that maybe it can be proven that the solution $(x,y)$ if it exists is bounded (via Lyapunov function or via first inegral), but I cannot find a solution yet.