Blackberry bushes are tough to keep under control. Left uncontrolled they will continue to invade your property. There are a number of approaches for control: mowing/mechanical control, herbicides, or using portable electric fence to for “target graze”. The ultimate goal is to remove the undesired weed and establish/leave desired species such as grass or other non-invasive bushes that will provided competition and ultimately suppress the undesired species from growing. But remember, any time you use pesticides always follow labels for proper use and application. There are non-chemical solutions that can be used as well like boiling water, or torching, though these techniques seldom kill the root system.
An IPM control approach that has been found to be fairly successful is using both mechanical and chemical control methods together. I feel attacking blackberry patches through the following steps will provide the best end result for you:
Spray with a broadleaf herbicide or glyphosate (non-selective, ie. kills all plants). This probably won’t be enough to kill all of the roots, but definitely stresses the plant. If you want to avoid harming the grass or lawn growing around the patch, use a broadleaf herbicide. If the berry patch is fairly large/thick and doesn’t have much grass growing around it, use the glyphosate. If the patch/thicket is too large to spray, mowing or grazing with goats, may be necessary as the first step, with a follow up herbicide application to the regrowth.
The best time to spray is early fall, right after the plant sets berries. After foliage dies, chip, shred, or mow over the stalks. This gets the plant down to a manageable height. Once the Blackberries are mowed down, either continue mowing, or spray with a broadleaf herbicide as you see new plants sprout. Establish a desired species for weed suppression for long term control. This process can take 3 or 4 years, but you should have things under control by then. Unfortunately, there is no easy solution.