Basically by control, he means if your at war, its part of your influence. In the case of a battle there you'd be the DEFENDER and they the attacker. Also, if someone is at war with you, you would be defending that system from them, much like SA defended California, Texas, and Colorado from AW/HF in the war, as it was in their influence.
If your a lawful faction, the influence/controlled system would be in your jurisdiction, allowing you to scan and demand cargo. Also anyone can demand to know a person's reasons for being in a system they control. For example, if a pirate came in Theta and I asked; Whats your business in Theta? And he answers; attacking traders/pirating. Thats when I would say we wouldn't allow it and he would either leave, fight, or try pirating and we would attack him.