Simply, a Fourier sine series is a sum of n sine waves over a given period. We can use this sum to approximate a given shape. For example, in lifting-line theory we approximate the lift distribution on a wing using a Fourier sine series.

I defined the altitude and velocity to be two Fourier sine series' of 20 coefficients, with an added linear component.

My method found that the best trajectory is a rounded trapezoid.

This shows that the best fuel burn would be to climb as quickly as possible to an altitude that minimizes fuel burn, and then nearly glide down to the destination airport.

This trajectory results in a fuel burn of 15980.28 lbf and a total flight time of 1.142 hrs.

The aircraft stays well below stall (CL ~= 1.4).

During the climbing phase, the aircraft burns fuel at a significantly higher rate than during the cruise or descent phases. This is to be expected: it would not be desirable to fly at full throttle when we want to descend!

The hills in the lift coefficient (and altitude and velocity) are an artifact of the Fourier sine series method. I assume these would level out with a greater number of Fourier terms.

As mentioned earlier, the aircraft flies at full throttle in the climb phase.

This steps back quite a bit during climb, and drops to zero during descent.

It is worth noting that we can see these points reflected in the thrust. The aircraft will hug the maximum throttle available once it has reached its desired climb angle during the climb phase.