My best guess is that your shrubby groundcover is a type of Ceanothus also called California lilac - but this plant is not a member of the lilac family. It can be controlled by cutting the woody stems now then raking up the branches and foliage and smothering the roots with cardboard and a mulch. If plants have no sunlight the roots will starve out and die. In a year you can remove the cardboard and dig into the dead roots - at least that is the plan. Some plants  can go dormant during their year of darkness and sprout again. If so keep the cardboard and mulch on for another year or resort to using an herbicide such as Roundup to kill the plant roots and all. Keep growing, Marianne Binetti