I require a numerical program to generate a random equation for me with the requirements listed below. This curve is for a velocity profile of an object, must have positive velocity at all times and has limits on the maximum acceleration and deceleration. I desire to create a slightly different curve each time I generate one, still meeting all the requirements.

The requirements for the function, v(t) for 0 < t < t_max, is:

- global max < b
- global min = 0
- v(0) = v(t_max) = 0
- v ' (t)_max <= a
- v ' (t)_min >= c
- integral ( v(t) ) = d

The following two requirements, as seen on the image, add no new information to the problem:

- v ' (0) <= a
- v ' (t_max) >= c

An image with this information is shown below:

Any help on this problem would be appreciated. If it is very lengthy to explain how to do kindly point me in the right direction as I have some maths textbooks.

For the numerical program I plan to use either MATLAB or Mathematica.

Thanks again!