I would suggest spraying Finale Herbicide along the fence line. It will not move in the soil nearly as bad as a soil sterile will. It also cheaper per application even though it is a costly upfront cost. I would make sure to mix it with a surfactant (I use Drexel's Surfac 910) as it will help it stick better to the surface of the plant. I recently purchased some Finale off of http://lawnandpestcontrolsupply.com
and have head excellent results fairly quickly. It was $199 but it will last a long time.