Difference between revisions of "Goddart's rocket problem"

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== Reference Solution ==
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The following reference solution was generated using BOCOP. The optimal value of the objective function is -0.63389.
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<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
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Image:Goddartacc.png| Control u over time.
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Image:Goddartpos.png| Position r over time.
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Image:Goddartspeed.png| Speed v over time.
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Image:Goddartmass.png| Mass m over time.
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</gallery>

Revision as of 15:27, 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}

Reference Solution

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


  • Reference solution plots

  • Control u over time.

  • Position r over time.

  • Speed v over time.

  • Mass m over time.

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