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Unformatted text preview: 1 = (1 − η )vt + h(ft − bt ), t = 0, 1, . . . , where η ∈ (0, 1) is a friction or drag parameter, ft is the engine thrust, and bt is the braking force,
at time period t. These must satisfy
0 ≤ bt ≤ min{B max , ft }, 0 ≤ ft ≤ F max ,
122 t = 0, 1, . . . , as well as a constraint on how fast the engine thrust can be changed,
ft+1 − ft  ≤ S, t = 0, 1, . . . . Here B max , F max , and S are given parameters. The initial thrust is f0 = 0. The takeoﬀ time is
T to = min{t  vt ≥ V to }, where V to is a given takeoﬀ velocity. The takeoﬀ position is P to = pT to ,
the position of the aircraft at the takeoﬀ time. The length of the runway is L > 0, so we must
have P to ≤ L.
(a) Explain how to ﬁnd the thrust and braking proﬁles that minimize the takeoﬀ time T to ,
respecting all constraints. Your solution can involve solving more than one convex problem,
if necessary.
(b) Solve the quickest takeoﬀ problem with data
h = 1, η = 0.05, B max = 0.5, F max...
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 Fall '13
 F.Borrelli
 The Aeneid

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