Difference between revisions of "Goddart's rocket problem"

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(Created page with "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...")
 
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A &=& 310 \\
 
A &=& 310 \\
 
k &=& 500 \\
 
k &=& 500 \\
 +
C &=& 0.6
 
\end{array}
 
\end{array}
 
</math>
 
</math>
 
</dd>
 
</dd>

Revision as of 14:20, 14 January 2016

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 \max_{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}

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