Everything (that has mass or energy) changes the shape of spacetime.

Sure, my point is that the changes made with Alcubierre drive are controllable.

Not really. The Alcubierre "drive" is closer to a "shape", a static solution to Einstein's equations. There is no proposed device that actually achieves this shape. The only idea along those lines implies a kind of Alcubierre railroad - you set up a path of negative mass matter (which almost certainly doesn't exist) and then move this with regular means as the ship is approaching, thereby continuing the bubble for the ship.