A simple Goddard problem

Goddard problem


This well-known problem (see for instance [1],[2]) models the ascent of a rocket through the atmosphere, and we restrict here ourselves to  vertical (monodimensional) trajectories.
The state variables are the altitude, speed and mass of the rocket during the flight, for a total dimension of 3. The rocket is subject to gravity, thrust and drag forces. The final time is free, and the objective is to reach a certain altitude with a minimal fuel consumption, ie a maximal final mass. All units are renormalized.

Latex formula

The drag is Latex formula, with the volumic mass is Latex formula.
We use the parameters Latex formula.

The Hamiltonian is an affine function of the control, so singular arcs may occur. We consider here a path constraint limiting the value of the drag effect: Latex formula. We see that depending on the value of C, the control structure changes. In the unconstrained case, the optimal trajectory presents a singular arc with a non-maximal thrust. When C is set under the maximal value attained by the drag in the unconstrained case, a constrained arc appears. If C is small enough, the singular arc is completely replaced by the constrained arc.

Numerical simulations:
Discretization: Gauss 4th order with 1000 steps.
Depending on the value of C, the optimal control structure changes from Bang-Singular-Bang to Bang-Constrained-Bang.



[1] R.H. Goddard. A Method of Reaching Extreme Altitudes, volume 71(2) of Smithsonian Miscellaneous Collections. Smithsonian institution, City of Washington, 1919.

[2] H. Seywald and E.M. Cliff. Goddard problem in presence of a dynamic pressure limit. Journal of Guidance, Control, and Dynamics, 16(4):776–781, 1993.

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