I have two optimisation problems.
In the first, I have a set number of equal time steps.
There are three state variables, longitude, latitude and mass. I integrate a dynamical system forward involving these. The heading angle is a control variable. The objective function is based on fuel burn rate, so that the aim is to make a journey burning as little fuel as possible. The final destination is a circular target set as an inequality constraint.
Fmincon is used to minimise the objective function given the inequality constraint. I keep the speed constant throughout each trajectory. The initial headings are taken as a set of headings round the Great Circle Route.
In the second version, I do much the same, but this time the speed is used as a second control. Initially the speed is set to a single value.
For some reason, my result for the single control uses less fuel, than my result from the two control problem, by a long way. I would have expected having the two controls would mean that I got at least the same as with one control if not better. The speeds that work best for the single control are about 20m/s slower than the average of the speed vector for the two control solution. Looking at the trajectories, I can see that they all 'work' in that they arrive at the right place and take the right time. It is just that higher speeds make the fuel burn rate higher, so more fuel is used for faster trips.
Can anyone think of any reasons why I might be getting these results please?