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Goddard Problem

Posed by R.H. Goddard, AMethod of Reaching Extreme Altitudes, Smithsonian Inst. Misc. Coll. 71, 1919, reprinted by Am. Rocket Soc., 1946.8

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Three state variablesr : distance from Earths center v : radial velocity m : rockets mass

One control variable : massow-rate8

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The dynamical modelr = v D(r, v) c 2 v = m m r m = with [0, max ].8

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The initial conditions are:r(0) = Re Earths radius v(0) = 0 start from rest m(0) = M0 initial mass

The (only) speciedend-condition is 1(x(tf )) = m(tf ) Mf8

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where Mf < M0 is the mass with all fuel expended

The cost functional isg(r(tf ), v(tf ), m(tf )) = r(tf ) , that is, we maximize the nal altitude.8

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Nondimensional Form

Its convenient tonondimenionalize. We select M : M0 unit mass L : Re unit length T :8

3 Re / unit time

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Note that this leads to /Re : unit speed2 M0/Re

: unit force

The scaled dynamical model r = v D(, v ) r c 1 v = 2 m m r m =

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with [0, max ], where, for example, r = r/Re

In the following we drop theand note that all quantities have been non-dimensionalized.8

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Applying the M.P.

We form the variational

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Hamiltonian H = r v + v c m m m D(r, v) 1 r2a

The adjoint dierential8

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equations are m r r3 v D v = r + m v v [c D(r, v)] . m = m2 The terminal transversality8

r = v

1D

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conditions imply H(tf ) = 0 r (tf ) = 0 v (tf ) = 0 m(tf ) = 1 Note that H is constant along an extremal path.8

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Applying the M. P. min H

We are to minimize thevariational Hamiltonian H subject to the bounds.

Observe that H can be written8

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as H = v c m + terms independent of m

Use the symbol S for the terms in square brackets

For the mass ow-rate we nd8

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three possibilities 0 = max singular

if S > 0 S