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.

Variables

The state variables $r,v,m$ describe the altitude(radius), speed and mass respectively.

The drag is given by

$D(r,v):=A{v}^2\rho (r)\text{, with }\rho (r):=exp(-k\cdot (r-r_0)).$

All units are renormalized.

Mathematical formulation

$\begin{array}{ll}\underset{m,r,v,u,T}{\min }& -m(T)\\ \text{s.t.}& \dot{r}(t)& =& v,\\ & \dot{v}(t)& =& -\frac{1}{r(t{)}^2}+\frac{1}{m(t)}(T_{max}u(t)-D(r,v))\\ & \dot{m}(t)& =& -b{T}_{max}u(t),\\ & u(t)& \in & [0,1]\\ & r(0)& =& r_0,\\ & v(0)& =& v_0,\\ & m(0)& =& m_0,\\ & r(T)& =& r_T,\\ & D(r(t),v(t))& \le & C\\ & Tfree\end{array}$

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.

Source Code

Model descriptions are available in:

• Bocop code at Goddart's rocket problem (Bocop)

References

The Problem can be found in the BOCOP User Guide.