View source for Goddart's rocket problem

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.
[[Category:MIOCP]]
== Variables ==
The state variables <math>r,v,m</math> describe the altitude(radius), speed and mass respectively.
 
The drag is given by 
<dd>
<math>
D(r,v):= Av^2 \rho(r)\text{, with }\rho(r):= exp(-k\cdot (r-r_0)).
</math>
</dd>
All units are renormalized.

== Mathematical formulation ==

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

== Parameters ==

<dd>
<math>
\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}
</math>
</dd>

== Reference Solution ==

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


<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 Image:Goddartacc.png| Control u over time.
 Image:Goddartpos.png| Position r over time.
 Image:Goddartspeed.png| Speed v over time.
 Image:Goddartmass.png| Mass m over time.
</gallery>

== Source Code ==
Model descriptions are available in:

* [[:Category: Bocop | Bocop code]] at [[Goddart's rocket problem (Bocop)]]


== References == 
The Problem can be found  in the  [http://bocop.org/ BOCOP User Guide].

[[Category:Aeronautics]]
[[Category:Minimum energy]]
[[Category:ODE model]]
[[Category:Path-constrained arcs]]
[[Category:Sensitivity-seeking arcs]]