The problems here are taken from examples in R.F. Stengel, "Optimal Control and Estimization", Dover.
The first four problems involve moving a resting cart 10m in 10s using as little total applied acceleration as possible.
In ex3.3.1a.mod, the acceleration is assumed to be constant. The constraint on final position is replaced with a quadratic penalty term in the objective function.
In ex3.3.1b.mod, the acceleration is assumed to vary linearly with time.
In ex3.3.1c.mod, the final velocity is not required to be zero. Instead a quadratic penalty term is added to the objective function.
Ex3.4.1.mod is a variant of ex3.3.1b.mod. In it, no assumption on the form of the acceleration is made. It is discovered that the assumption of linearity in ex3.3.1b.mod was in fact optimal.