For the double integrator with a discontinuous coefficient at the control, we obtain an attraction set—an asymptotic variant of reachable sets corresponding to constraints of asymptotic character—and study the latter’s properties. 
These constraints correspond to a control mode in the class of short-time pulses. An additional requirement is to fully consume the available energy resources. To calculate the attraction set numerically, an algorithm was developed and implemented. The results of computational experiments are presented.