Then, its your choice, how you want to do it... first, you should treat the speed of the animation as a variable. Then, you should choose how you want the behaviour to be. You could define a mathematical function that mimics the behaviour you want in the style of:
animation_speed = animation_speed + any_function(animation_speed) * delta_simt.
The simplest example of such a function would just be the classic arithmetic average function.
any_function = (animation_speed + target_speed) / 2
You could also use a state machine. Lets say, you have four states: Off, spool up, run, spool down. For starting the animation, you go from off state to spool up. In spool up, you just increase the animation_speed until it has reached your reference speed. When the reference speed is exceeded, the state switches to run. When you want to turn the animation off, you leave run towards the state spool down and decrease your animation speed until its zero or less. Then it sets the animation speed to zero and goes to off state again.