2003 Cadillac CTS Keyless Door Lock

Other Category problem
2003 Cadillac CTS V6 Two Wheel Drive Automatic 35000 miles

My wife's 2003 CTS has a keyless door lock problem. When you turn off the car and get out you can lock, and unlock, the doors with the keyless remote. But, after a very short period of time the car no longer responds to the remote. So, if you lock it with the remote, walk away and come back a few minutes later, it does not respond to the keyless entry remote. It's as if there is some circuit that powers down after a short interval. The motes have new batteries and obvisously work since they will lock, and unlock, the doors at the time the car is turned off. Anyone seen something like this and have a solution?