Goddart's rocket problem

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In Goddart's rocket problem we model the ascent (vertical; restricted to 1 dimension) of a rocket. The aim is to reach a certain altitude with minimal fuel consumption. It is equivalent to maximize the mass at the final altitude.

Variables

The state variables r,v,m describe the altitude, speed and mass.

The drag is given by


D(r,v):= Av^2 \rho(r)\text{, with }\rho(r):= exp(-k\cdot (r-r_0)).
All units are renormalized.

Mathematical formulation


\begin{array}{llcll}
 \displaystyle \min_{m,r,v,u,T} &  -m(T)\\[1.5ex]
 \mbox{s.t.} & \dot{r} & = & v, \\
 & \dot{v} & = & -\frac{1}{r^2} + \frac{1}{m} (T_{max}u-D(r,v)) \\[1.5ex]
& \dot{m} & = & -b T_{max} u, \\
& u(\cdot) &\in& [0,1] \\
 & r(0) &=& r_0, \\
 & v(0) &=& v_0, \\
 & m(0) &=& m_0, \\
 & r(T) &=& r_T, \\
 & D(r(\cdot),v(\cdot))&\le& C \\
& T free
\end{array}

Parameters


\begin{array}{rcl}
r_0 &=& 1  \\
v_0 &=& 0  \\
m_0 &=& 1  \\
r_T &=& 1.01 \\
b &=& 7 \\
T_{max} &=& 3.5 \\
A &=& 310 \\
k &=& 500 \\
C &=& 0.6
\end{array}

Reference Solution

The following reference solution was generated using BOCOP. The optimal value of the objective function is -0.63389.


References

The Problem can be found in the BOCOP User Guide.