Is your TRACTION CONTROL and ABS working properly? Sounds like it may be a bad wheel speed sensor to me since it turns on but turns itself off pretty quick.
Has the car been MODDED any and TUNED? Does it have stock 17"wheels and stock tire sizes? Wheel/Tire size has to be set in the PCM for all the stuff to work together.
Check for any fault codes first. The cruise control is tied to several different systems and won't work unless all the other systems and components are working. Wheel speed sensors on left and right sides of the rear diff, cruise control module/actuator under the hood next to the ABS valve block. Has the BCM been changed out? If so you will need a TECH II to reflash it to match your VIN, GM car dealerships and most good repair shops have the TECH II tool.
A low battery can also cause multiple electronic gremlins as well as the wiring close to the BCM under the dash in the passenger footwell behind the glovebox, check to make sure the wiring isn't rubbed/chaffed.
You can check for any stored fault codes by putting the KEY into the ignition, hold the MODE and SET buttons down, turn the KEY to ACC (turn key on without starting engine), use the UP/DOWN buttons to scroll through the menu items until you get to FAULT CODES, see what codes are stored, write them down and google them to see what they are.
Drive down to your local Autozone, O'Rileys or whatever place you have local that has a scan tool and have them connect it to your ODBII port under the steering wheel, pull up the codes and CLEAR/DELETE them. The store employees can't/aren't allowed to delete them but you can.