Mike a minute
Here are your options:
Go to the links at the bottom and click on "Weeds Gone Wild" for the most thorough information.
Because this is an annual and has to come up again from seed each year, don't let it produce berries and drop them for next year's vines.
After you kill the vines or remove them, cover the area with a smothering mulch that will prevent seeds from sprouting next year. Several layers of newspapers, overlapping, spread under the mulch will make an impenetrable barrier to an seed germinating. Do not use plastic.
Nature hates a vacuum. If you have bare areas on that hill, fill them with plants or mulch at all times.