In optimal control, if the Hamiltonian $H$ is linear in the control $u$, then the optimality condition
$$\frac{\partial H}{\partial u}$$
gives no information about the optimal control $u^*$. The way to go about (see this) it is to define the switching function
$$\phi(t) = \frac{\partial H}{\partial u}$$
and show that if the control is not singular, then it must be a bang-bang control. To determine the switching times $t_S$ of this bang-bang control, we usually look at the number if times $\phi(t)$ changes signs. Thus If $\phi$ is linear, we can say that there is at most one switching time.
My question is: How do we systematically prove that there is at most one (or n number) of switching time if the switching function $\phi(t) = F(x(t))$ is some nonlinear function? This of course must take into account arbitrary initial conditions
In general this is a difficult problem and case-by-case analysis is usually necessary as you will need to take into account the optimal trajectory explicitly for every possible initial condition. As almost all nonlinear systems do not admit any closed-form solution, this approach is unlikely to be applicable.
A possible workaround would be to show that the switching function is monotonic. In such a case, this would imply that there is at most one switching.
Also, when you say that $\phi(t)$ is linear, the linearity is with respect to what? I am not so sure that your conclusion is correct in this situation.