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For a landing via suicide burn, I have identified three phases:

  1. Starting at $t_0$, the craft approaches the ground with $v_x$ constant and $v_y$ gradually increasing (due to gravity)
  2. At $t_b$, the craft accelerates in retrograde direction until the speed vector $(v_x,v_y)$ is below a certain safety margin.
  3. The craft maintains the speed inside the safety margin until it touches down at $t_1$

I am trying to optimize $t_b$ for fuel consumption.

Obviously, there are three cases to consider:

  1. $t_b$ might be too early, causing the fuel to run out and the craft to crash
  2. $t_b$ might be too late, causing the craft to crash due to a high residual velocity
  3. $t_b$ might be inside a safe landing corridor

In order to optimize this, I need a goal function, i.e. some kind of score that I assign to each outcome. The case 3 is trivial: Here we can attempt to maximize the residual fuel when landed. In case 2, I can probably minimize the residual velocity.

But what about case 1? Intuitively, if the lander has enough power to brake to a standstill (and something), residual velocity should be smaller for larger $t_b$, but is that true?

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  • $\begingroup$ Isn't the whole point of a suicide burn that you hit $v_x=0, v_y=0$ at exactly the moment of touchdown? At that point, you don't need any additional fuel. So for a suicide burn, you'd optimize for minimum residual fuel when your velocity and altitude are both zero. $\endgroup$
    – user
    Commented May 2, 2017 at 20:11
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    $\begingroup$ @MichaelKjörling That's fine for idealized mathematical solutions, but in practice you want to end the maximum-thrust suicide burn phase with some downward velocity at some positive altitude with enough remaining fuel to manage the rest of the descent incrementally. $\endgroup$ Commented May 2, 2017 at 20:20
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    $\begingroup$ The optimum case is trivial--burn at full throttle until the engines shut down due to starvation at the instant you touch the target. In practice you will back off from this a bit as none of the factors involved can be measured with extreme precision. $\endgroup$ Commented May 2, 2017 at 20:32
  • $\begingroup$ What you are describing is not a suicide burn it's a landing. The most fuel-efficient landing is the suicide burn. $\endgroup$ Commented May 2, 2017 at 20:53
  • $\begingroup$ @LorenPechtel I wouldn't call starving the engines optimal, since it could be preferred to land with as much remaining fuel (for a return trip perhaps?) as possible. Nor would I call finding time of ignition for this kind of burn a trivial task, as it pretty much boils down to the problem of optimizing a powered trajectory under inverse-square gravity field. $\endgroup$
    – Przemek D
    Commented May 10, 2017 at 13:27

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