Rear Propulsion Steering Spline, or what is this called?

Question

What I am looking for is essentially to know what this is, or what I can google to learn how to do it... unless of course the solution is simple.

You have an object at the start position facing a direction, and a desired position and direction.

The object has:

max speed
rotation rate
acceleration
deceleration

I am essentially trying to simulate the object having a "thruster" force acting on it.

I would like a smooth path from VecStart to VecEnd, where the object accelerates from stationary, and then decelerates as it gets to its desired position and rotation.

I am at a loss for what this is called, and where I should begin to research.

An exact/optimal solution to the problem you pose is in the domain of Control Theory, and so far I've found it a bit too cumbersome to tackle precisely that way. (An engineer may do better here than a lowly game designer like me. ) But I do have a couple of previous answers looking at ways we can get decent solutions by thinking of our path as a Bézier spline. Quadradic for constant thrust, or cubic for linearly changing thrust. — DMGregory, Jul 23 '21 at 00:51
@DMGregory Wow, this is a great response. Thank you Douglas. Cubic bezier looks to me like the easiest way forward. — puttputt, Jul 23 '21 at 16:41

Rear Propulsion Steering Spline, or what is this called?

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