Proof request for Optimal Control of : $\dot{x} = Ax + ub, \; |u(\cdot)| \leq 1$

Question

I am kindly requestion for an elaborative proof of the following Lemma I encountered during my elementary studies of Optimal Control :

Lemma : For the controllable system $\dot{x} = Ax + ub, \; |u(\cdot)| \leq 1, \; x_0 \neq 0$ if the control $u:[0,t^*]\to [-1,1]$ is optimal, then it satisfies the Maximum Principle.

Thanks in advance for any elaborations or links to any available proof.