🚀 Rocket Landing

Land a returning booster softly on the pad with as much fuel left over as possible. You're picking a piecewise-constant throttle schedule — 12 values in [0, 1], one per 0.83-second segment. The engine has a thrust-to-weight ratio of ~4.8 and a fuel tank good for 4 seconds of full burn — so the rocket can dump fuel fast enough to do a real suicide burn: free-fall most of the way, then one precisely-timed full-throttle segment that brings v ≈ 0 at h = 0. Free fall crashes you in at 65 px/s; constant full throttle wastes everything and sends you back to space; the right answer is a near-bang-bang schedule with a fractional kicker that fakes the sub-segment timing.

What's happening

The booster starts at 200 px altitude with 15 px/s downward velocity. Constant gravity g = 10 pulls it down; a single throttle τ ∈ [0, 1] controls a thrust producing acceleration 48·τ upward (so τ ≈ 0.21 is the hover throttle and full thrust decelerates a falling rocket at 38 px/s²). The 12 control variables are the throttle settings for 12 equal time-segments spanning 10 seconds. Fuel burns at full throttle in exactly 4 seconds — so the rocket has plenty for a short suicide burn but constant max throttle empties the tank before landing.

Score rewards landing softly and keeping fuel. Touchdown velocity v gives 100 · (1 − v/80) on any landing (a perfect v ≈ 0 scores 100; the rocket vaporises in a fireball above 5 px/s but still earns partial credit down to 0 at v = 80) plus up to +10 for fuel left in the tank. A schedule that never lands within 10 seconds is scored on how close it came — the gradient toward soft landings stays positive even from the airborne region, so optimizers aren't trapped at "free fall" as a flat local optimum.

The landscape has several distinct local optima — a gradual "hover lower and lower" descent, a linear taper, and the razor-thin suicide burn where the rocket free-falls until exactly the right instant then goes to full thrust. Trust-region methods often get caught between strategies. Differential Evolution and CMA-ES usually find the suicide-burn region.

Simplified 1-D vertical landing. Real boosters add gimbal control, mass-varying centre of gravity, atmospheric drag, and aerodynamic surfaces — but the optimal-control structure (and the suicide-burn solution) is the same.

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