Pontryagin principle, Optimal control or Numerical scheme ? logical constraint?

Question

I have the following optimal control problem :

$ \textit{ Minimize } \qquad J(u,x,T) =\displaystyle \int_0^T \Big[ u_1(t)\log\frac{u_1(t)}{ x_1(t)x_2(t)} - u_1(t) +x_1(t)x_2(t) \qquad \qquad \qquad \quad \qquad \qquad \qquad\qquad\qquad\qquad + \: u_2(t)\log\frac{u_2(t)}{x_2(t)} - u_2(t) + x_2(t) \Big] dt$

subject to the dynamic $\displaystyle \qquad \left[\begin{array}{l} \dot{x}_1(t)\\ \dot{x}_2(t) \end{array}\right] = \left[\begin{array}{rcl} -u_1(t)&+&u_2(t)\\ u_1(t)&& \end{array}\right]$,

with the constraints

1. $T>0$ is free

2. $x_1(0), x_2(0)$ are fixed, $x_2(T) =0$ but $x_1(T)$ is free.

3. $x_1(t),x_2(t)\geq 0$

4. $u_1(t),u_2(t)\geq 0$

5. $u_1(t) >0 $ only if $x_2(t)x_1(t)>0$

6. $u_2(t) >0 $ only if $x_2(t)>0$

How can we handle constraints $5.$ and $6.$ ? it seems to be logical constraints?

Pontryangin principle could work for this problem? considering constraints $5.$ and $6.$ ?

I didn't found a lecture note or Book on this kind of problem.

I also want to solve it Numerically, could any one help with a reference?

Thanks in advance!!!