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.

The drag is , with the volumic mass is .

We use the parameters .

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: . 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***.
*

**References**

[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.