LAUSR

A new concept called backward controller for an evolution process is introduced, which is the minimal compact set controlling all fibers of the pullback attractor in the past. The existence criteria are established by using the backward flattening compactness and backward equi-continuity of the system if the phase space is all of continuous vector-valued functions. Both backward controller and backward compact attractor are obtained for the non-autonomous Benjamin-Bona-Mahony equation with variable or distributed delay.